Character mistake: When Maggie starts speaking Ogdenvillian, Marge says that it is her first word. But in the episode "Gump Roast", everyone, the Simpsons included, all watched a video of Maggie saying her first word. (00:17:35)
Other mistake: When Bart is looking at Doctor Hibbert on the rowing machine, all the pictures on the walls behind him don't have faces on them.
Another Simpson Clip Show - S6-E3
Character mistake: In the first scene, Marge says that it is 9:30pm, but after Homer's flashback, the clock next to Homer reads 11:50.
Homer and Lisa Exchange Cross Words - S20-E6
Plot hole: Gil would have been disqualified for cheating at the Crossword Championship. (00:15:00)
Plot hole: When Bart locks Principal Skinner and Mrs Krabappel in Skinner's office, and tells them that they can't come out until they've reached an agreement, why didn't Skinner and Krabappel just climb out Skinner's office window instead of spending hours pounding on the door yelling to be let out?
Please Homer, Don't Hammer 'Em - S18-E3
Continuity mistake: In this episode Principal Skinner breaks into Bart's doctors office to find out if Bart has a 'kryptonite'. Bart discovered Skinner was allergic to peanuts. On Bart's medical history form under 'known allergies' it has shrimp ticked (they later battle with shrimp and peanut on a stick). However in a previous episode (The Heartbroke Kid, season 16 episode 17) it was discovered that he was allergic to cauliflower, so that should have been on the list as well.
Don't Fear The Roofer - S16-E16
Revealing mistake: When Homer sits on Lenny's cake in Moe's bar, some of the frosting appears to stay in mid air, and when the frosting is on the wall, it doesn't slide to the ground like frosting would.
Continuity mistake: Millhouse's hair color changes from when we first see him to when Bart comes back to the school yard.
Continuity mistake: In the surgeon's office there are three certificates on the wall with ribbons on them. In the first shot there are two red ribbons and a blue one but in the shot where Homer stands up to read the bill all the ribbons are red.
Continuity mistake: After the boy's car gets crushed, Bart sells Milhouse's glasses for $600. Then Milhouse knocks over the Coke bottles and the clerk gets the money back and keeps the glasses. The next scene, where they try to get on a train that goes to Springfield, you can see Milhouse in his glasses and throughout the rest of the episode.
The Dad Who Knew Too Little - S14-E8
Other mistake: When Bart flashes Lisa's laser at Principle Skinner's bottom, Chief Wiggum can still see it even when his car is in the way.
Guess Who's Coming to Criticize Dinner - S11-E3
Other mistake: In the German version, the Italian cook peeved by Homer's comment on lasagna swings the cleaver at him using an Italian word ("Porcodio") which is a rather strong blasphemy, way stronger than any other expression used in original or in other localizations worldwide.
Character mistake: When Chief Wiggum shows up to arrest Nelson during the game at the end of the episode, Bart goes with Wiggum instead while pretending to be Nelson. Even a cop as incompetent as Chief Wiggum would not have mistaken Bart for Nelson since he'd had numerous dealings with both before this episode; Wiggum may not be particularly bright, but he knows who's who in Springfield.
Factual error: When Bart is at Nelson's house and Nelson is firing the BB gun, bullets should come out of his gun when he pumps it. None ever come out, and several times he is shown firing without pumping.
I Don't Wanna Know Why the Caged Bird Sings - S19-E4
Factual error: During Dwight's attempted bank robbery, he gets hit with a couple of dye packs, then appears a few shots later totally clean. The dye packs banks use are notoriously difficult to wash off once they're triggered; even if Dwight did manage to clean himself up, there would still be remnants of dye on him.
Simpsoncalifragilisticexpiala(Annoyed Grunt)cious - S8-E13
Audio problem: When Shary Bobbins suggests to Lisa that they could have a tea party on the ceiling, Shary's lips don't match what she's saying.
Character mistake: When Mr. Burns and Smithers are walking through the morgue, the sign on the wall is incorrectly spelled "Hospitial Morgue."
The Simpsons Christmas Special: Simpsons Roasting on an Open Fire - S1-E1
Audio problem: When Lisa and Maggie are touching Bart's surgically removed tattoo, he says "Ow", but look at the way he says it. His mouth moves when he says "O" but doesn't move when he says "w". (00:07:50)
Homer's Barbershop Quartet - S5-E1
Continuity mistake: When Bart and Lisa find the record Melvin and the Squirrels, they hand it to the Comic Book Guy to play. While the record is in his hand you can still see it displayed on the rack.