What I find, and what IMDB has to say.
Pilot:
Continuity: When the Chief asks Lucinda to tell Shawn to leave, she opens and closes the door to the Chief's office, but in the next shot, when Lassiter comes out, the door is open.
Continuity: During Gus's introduction, when Shawn walks into Gus's office, the door is open and closed between shots.
Plot Holes: When Shawn is screaming about the Consumine, he yells out a doctor's name, Dr. Bindali. Gus, however, never told Shawn a doctor's name, simply the name of the drug.
Inproper Protocol: In the International Pilot, when Shawn comes walking into the department after solving the case, the officer who believes in psychics is on the phone. She is clearly heard asking someone what the color of their car is. Not even five seconds later, upon seeing Shawn, she hangs up on the distressed caller.
Spellingg Bee:
Factual errors: While Gus is looking through the pamphlet, he notes the words that won in 1985 and 1943. However, there was no spelling bee between 1943 and 1945 during WWII.
Revealing mistakes: When Gus leaves the courthouse to start investigating, the door opens before he even reaches it, and continues to stay open until after Shawn passes through the door as well.
Speak Now or Forever Hold Your Piece:
Woman Seeking Dead Husband; Smokers Okay, No Pets:
Continuity: Shawn gets up twice from the couch whilst chatting with the bank robbers.
9 Lives:
Errors made by characters (possibly deliberate errors by the filmmakers): Shawn repeatedly refers to the cat as "the little boy cat." The cat is calico, a color pattern that occurs only in females. The filmmakers were fully aware the cat was a female, however James Roday (Shawn) thought it would be funnier if the cat was thought to be male.
Anachronisms: When young Shawn (which takes place in the mid 1980's) is looking for the decoder ring, Henry tells Shawn he can get the 411 on the cereal company. 411 was not used in the 1980's.
Revealing mistakes: When a forensics investigator is kneeling beside the dead body on the couch, in the first crime scene, the "corpse" is very clearly breathing.
Weekend Warriors:
Miscellaneous: When Lassiter arrests the dentist, he says "You're under the arrest for the murder of Nelson Poe." He should have just said "under arrest" but the flaw wasn't caught and the scene was left in.
Who You Gonna Call?:
Audio/visual unsynchronized: The scene where Shawn and Gus are checking out the broken mirror and Shawn touches it, Gus (when the camera is showing his side/back) says something like "Enjoy your Hepatitis" but Gus' Mouth Never moves.
Shawn vs The Red Phantom:
Continuity: When Shawn takes the final note, "Judgment Day," off the woman's' heel, there is clearly a hole in the note. But when it cuts to Shawn's point of view the hole is gone.
Forget Me Not:
From The Earth to The Starbucks:
Continuity: In Tom Blair's Pub at the beginning of the episode, Shawn very clearly takes the lime off the rim of his drink and drops it into the glass. When he goes to stand and leave the counter, the lime is back on the rim of the glass. Then, as he turns, it's back inside the glass again.
He Loves Me, He Loves Me Not, He Loves Me, Oops, He's Dead:
Errors made by characters (possibly deliberate errors by the filmmakers): Shawn says about one of the potential suspects at the speed dating, "...she drives an 04 Trans Am..." General Motors stopped making Pontiac Firebirds/Trans Ams in 2002.
Miscellaneous: The original transmission of this episode used the end credits from the previous week's "From the Earth to Starbucks".
Continuity: When Shawn is speaking in the chief's office about the speed-dating, there is no flower on his shirt before the "commercial break". After the break, he has a red flower attached to his shirt.
Cloudy... With a Chance of Murder:
Continuity: Juliet claims she is wearing alligator pumps as she and Lassiter are walking up to the courthouse, but a subsequent closeup shows she is not.
Continuity: Lassiter is already sitting in the court room crowd before walking into the building.
Game, Set, Muuurder?:
Poker? I Barely Know Her:
Continuity: When Shawn is trying to read the cards on his floor, he grabs a jack of hearts, but when the camera shifts back to the floor it is still there.
Scary Sherry: Bianca's Toast:
Continuity: In Henry's flashback of him grabbing Scary Sherry and pulling her out of the window, one of the windows is pulled inside. However, in the beginning of the episode, after Shawn and Gus think she has jumped, both windows were wide open.
Continuity: In the end of the episode, Gus is forgotten by Shawn, Lassiter and O'Hara inside Wispy Sunny Pines, although, according to the scenes of them entering the building, they had to pass through him in order to get out.
Continuity: When about to enter Wispy Sunny Pines, O'Hara puts the phone in her bag and the bag on her shoulder, but after the camera shot shifts the bag is on her elbow.
Continuity: When dining with Shawn and Gus, between shots, Henry is holding his beer by the middle of the bottle, then by the neck and then by the middle again.
Factual errors: The BKO Sorotity Annual Makeover Fund Raiser wants to "save the endangered Tamarind". The Tamarind is actually a tree native to eastern Africa which has been introduced into most worldwide tropical regions and is not in danger whatsoever. What they truly wanted to save is the Tamarin, a name used to any squirrel-sized New World monkeys from the family Cebidae.
Continuity: In the beginning of the episode, when O'Hara is talking with Shawn and Gus, she removes her pink paste off the table with both hands, but when the camera shifts, she does it again but with only one hand this time.
American Duos:
Continuity: The girls in the hall and the hip hop dancers on stage while Shawn and Gus were on the catwalk, both shared the same contestant number 43101.
65 Million Years Off:
Continuity: While parking in front of a gate of the Deacon Walker Farms, you can perfectly see between shots Gus isn't wearing a seatbelt, then he is, then he isn't again, then finally he is when he leaves the car.
Psy vs Psy:
Continuity: At the beginning Special Agent Ewing comments that they must have missed the suspect by a few minutes. However, at the end during the flashback, this changes to five minutes.
Continuity: When Shawn kicks over Lindsay's suitcase, he topples it onto its front (the side with the zippers), but when the camera changes angles, the suitcase has flipped onto its back.
Errors made by characters (possibly deliberate errors by the filmmakers): When Lassiter is commenting on Mildred, he states that it must be an anagram. The proper term is acronym. An anagram is when you mix up the letters of a word to make another word, and an acronym is a group of letters that each stand for a different word.
Errors made by characters (possibly deliberate errors by the filmmakers): Shawn claims that 700 hours is almost 20 days but it is almost 30 days.
Continuity: When Shawn is outside the Psych office talking to Gus about how the other psychic is bothering him, there are two guys standing in the background leaning against the rail. In the next shot, there is only one guy, then it switches back to two guys in the very next shot.
Zero to Murder in Sixty Seconds:
And Down the Stretch Comes Murder:
Errors made by characters (possibly deliberate errors by the filmmakers): Shawn tells Gus that he wanted to be the Milwaukee Brewers' mascot, a giant sausage, when he grew up. The Brewers' mascot is actually Bernie Brewer, a blonde German man with a large mustache (although Milwaukee home games traditionally feature "giant sausage" races during the seventh inning.)
Meat is Murder, But Murder is Also Murder:
Continuity: When Shawn is trying to get a job at the Mirror, he impresses editor Phil Pritikin by deducing that the editor is left-handed. However, in several scenes, Pritikin can be seen writing with his right hand.
If You're So Smart, Then Why Are You Dead?:
Rob-a-bye Baby:
Factual errors: Shawn takes pictures of the crime scene photos on Lassiter's desk using his Motorola RAZR phone, pressing the same button rapidly to take several pictures in a row. However, this particular model of phone requires the user to save and name each picture individually before being allowed to take another; it would be impossible to take multiple shots so quickly.
Bounty Hunters!:
*I'll add more as I find them.
Pilot:
Continuity: When the Chief asks Lucinda to tell Shawn to leave, she opens and closes the door to the Chief's office, but in the next shot, when Lassiter comes out, the door is open.
Continuity: During Gus's introduction, when Shawn walks into Gus's office, the door is open and closed between shots.
Plot Holes: When Shawn is screaming about the Consumine, he yells out a doctor's name, Dr. Bindali. Gus, however, never told Shawn a doctor's name, simply the name of the drug.
Inproper Protocol: In the International Pilot, when Shawn comes walking into the department after solving the case, the officer who believes in psychics is on the phone. She is clearly heard asking someone what the color of their car is. Not even five seconds later, upon seeing Shawn, she hangs up on the distressed caller.
Spellingg Bee:
Factual errors: While Gus is looking through the pamphlet, he notes the words that won in 1985 and 1943. However, there was no spelling bee between 1943 and 1945 during WWII.
Revealing mistakes: When Gus leaves the courthouse to start investigating, the door opens before he even reaches it, and continues to stay open until after Shawn passes through the door as well.
Speak Now or Forever Hold Your Piece:
Woman Seeking Dead Husband; Smokers Okay, No Pets:
Continuity: Shawn gets up twice from the couch whilst chatting with the bank robbers.
9 Lives:
Errors made by characters (possibly deliberate errors by the filmmakers): Shawn repeatedly refers to the cat as "the little boy cat." The cat is calico, a color pattern that occurs only in females. The filmmakers were fully aware the cat was a female, however James Roday (Shawn) thought it would be funnier if the cat was thought to be male.
Anachronisms: When young Shawn (which takes place in the mid 1980's) is looking for the decoder ring, Henry tells Shawn he can get the 411 on the cereal company. 411 was not used in the 1980's.
Revealing mistakes: When a forensics investigator is kneeling beside the dead body on the couch, in the first crime scene, the "corpse" is very clearly breathing.
Weekend Warriors:
Miscellaneous: When Lassiter arrests the dentist, he says "You're under the arrest for the murder of Nelson Poe." He should have just said "under arrest" but the flaw wasn't caught and the scene was left in.
Who You Gonna Call?:
Audio/visual unsynchronized: The scene where Shawn and Gus are checking out the broken mirror and Shawn touches it, Gus (when the camera is showing his side/back) says something like "Enjoy your Hepatitis" but Gus' Mouth Never moves.
Shawn vs The Red Phantom:
Continuity: When Shawn takes the final note, "Judgment Day," off the woman's' heel, there is clearly a hole in the note. But when it cuts to Shawn's point of view the hole is gone.
Forget Me Not:
From The Earth to The Starbucks:
Continuity: In Tom Blair's Pub at the beginning of the episode, Shawn very clearly takes the lime off the rim of his drink and drops it into the glass. When he goes to stand and leave the counter, the lime is back on the rim of the glass. Then, as he turns, it's back inside the glass again.
He Loves Me, He Loves Me Not, He Loves Me, Oops, He's Dead:
Errors made by characters (possibly deliberate errors by the filmmakers): Shawn says about one of the potential suspects at the speed dating, "...she drives an 04 Trans Am..." General Motors stopped making Pontiac Firebirds/Trans Ams in 2002.
Miscellaneous: The original transmission of this episode used the end credits from the previous week's "From the Earth to Starbucks".
Continuity: When Shawn is speaking in the chief's office about the speed-dating, there is no flower on his shirt before the "commercial break". After the break, he has a red flower attached to his shirt.
Cloudy... With a Chance of Murder:
Continuity: Juliet claims she is wearing alligator pumps as she and Lassiter are walking up to the courthouse, but a subsequent closeup shows she is not.
Continuity: Lassiter is already sitting in the court room crowd before walking into the building.
Game, Set, Muuurder?:
Poker? I Barely Know Her:
Continuity: When Shawn is trying to read the cards on his floor, he grabs a jack of hearts, but when the camera shifts back to the floor it is still there.
Scary Sherry: Bianca's Toast:
Continuity: In Henry's flashback of him grabbing Scary Sherry and pulling her out of the window, one of the windows is pulled inside. However, in the beginning of the episode, after Shawn and Gus think she has jumped, both windows were wide open.
Continuity: In the end of the episode, Gus is forgotten by Shawn, Lassiter and O'Hara inside Wispy Sunny Pines, although, according to the scenes of them entering the building, they had to pass through him in order to get out.
Continuity: When about to enter Wispy Sunny Pines, O'Hara puts the phone in her bag and the bag on her shoulder, but after the camera shot shifts the bag is on her elbow.
Continuity: When dining with Shawn and Gus, between shots, Henry is holding his beer by the middle of the bottle, then by the neck and then by the middle again.
Factual errors: The BKO Sorotity Annual Makeover Fund Raiser wants to "save the endangered Tamarind". The Tamarind is actually a tree native to eastern Africa which has been introduced into most worldwide tropical regions and is not in danger whatsoever. What they truly wanted to save is the Tamarin, a name used to any squirrel-sized New World monkeys from the family Cebidae.
Continuity: In the beginning of the episode, when O'Hara is talking with Shawn and Gus, she removes her pink paste off the table with both hands, but when the camera shifts, she does it again but with only one hand this time.
American Duos:
Continuity: The girls in the hall and the hip hop dancers on stage while Shawn and Gus were on the catwalk, both shared the same contestant number 43101.
65 Million Years Off:
Continuity: While parking in front of a gate of the Deacon Walker Farms, you can perfectly see between shots Gus isn't wearing a seatbelt, then he is, then he isn't again, then finally he is when he leaves the car.
Psy vs Psy:
Continuity: At the beginning Special Agent Ewing comments that they must have missed the suspect by a few minutes. However, at the end during the flashback, this changes to five minutes.
Continuity: When Shawn kicks over Lindsay's suitcase, he topples it onto its front (the side with the zippers), but when the camera changes angles, the suitcase has flipped onto its back.
Errors made by characters (possibly deliberate errors by the filmmakers): When Lassiter is commenting on Mildred, he states that it must be an anagram. The proper term is acronym. An anagram is when you mix up the letters of a word to make another word, and an acronym is a group of letters that each stand for a different word.
Errors made by characters (possibly deliberate errors by the filmmakers): Shawn claims that 700 hours is almost 20 days but it is almost 30 days.
Continuity: When Shawn is outside the Psych office talking to Gus about how the other psychic is bothering him, there are two guys standing in the background leaning against the rail. In the next shot, there is only one guy, then it switches back to two guys in the very next shot.
Zero to Murder in Sixty Seconds:
And Down the Stretch Comes Murder:
Errors made by characters (possibly deliberate errors by the filmmakers): Shawn tells Gus that he wanted to be the Milwaukee Brewers' mascot, a giant sausage, when he grew up. The Brewers' mascot is actually Bernie Brewer, a blonde German man with a large mustache (although Milwaukee home games traditionally feature "giant sausage" races during the seventh inning.)
Meat is Murder, But Murder is Also Murder:
Continuity: When Shawn is trying to get a job at the Mirror, he impresses editor Phil Pritikin by deducing that the editor is left-handed. However, in several scenes, Pritikin can be seen writing with his right hand.
If You're So Smart, Then Why Are You Dead?:
Rob-a-bye Baby:
Factual errors: Shawn takes pictures of the crime scene photos on Lassiter's desk using his Motorola RAZR phone, pressing the same button rapidly to take several pictures in a row. However, this particular model of phone requires the user to save and name each picture individually before being allowed to take another; it would be impossible to take multiple shots so quickly.
Bounty Hunters!:
*I'll add more as I find them.
