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.
Love, Springfieldian Style - S19-E12
Continuity mistake: In the third story of the episode, Bart and Nelson throw Martin into the Jukebox despite it never being opened. When he is trapped inside the box the glass is firmly in place as well.
Marge vs. Singles, Seniors, Childless Couples and Teens, and Gays - S15-E8
Factual error: The symbols on the heads of the Teletubbies on stage are wrong. They did use the word 'teletubbies', so they're meant to be the same as the real world. (00:05:40)
Continuity mistake: Near the end when Homer is going into the show, he is not holding his coat. Then it appears in his hands and gets stolen. In the very next scene, he then wears his coat again. (00:19:40)