Continuity mistake: Before Homer and Marge get married, they are taking more steps then they are supposed to, as you can see the interior of the wedding chapel before this scene. (00:10:10)

Burns Verkaufen der Kraftwerk - S3-E11
Continuity mistake: At the beginning, Mr. Burns pats his alligator, and his hand is instantly down in the next shot. (00:01:50)

Continuity mistake: Patty sprays Barney with pepper spray. It disappears from her hand in the next shot. (00:18:55)
Continuity mistake: At the end, when Bart gets his trophy, Marge and Homer switch positions. (00:20:35)
Continuity mistake: Homer drives to Moe's Tavern with his normal shirt on. He has his green jacket on top of it in the next shot. (00:04:45)

Lisa's Pony - S3-E8
Continuity mistake: Homer writes '4 1/2 Reed' on his left shoe. When the owner of the music store asks him what Lisa plays, he is reading off his right shoe. (00:02:00 - 00:04:40)
Lisa's Pony - S3-E8
Continuity mistake: In the last shot of Marge and Homer's discussion of buying a pony, we can see a door on the right side of the screen. This door changes position when Lisa rides in the bedroom with her pony. (00:08:45 - 00:12:10)
Audio problem: At the Penitentiary gates when Ralph says 'Daddy's key went in there,' his lips don't move.
Who Shot Mr. Burns? (1) - S6-E25
Continuity mistake: When Smithers and Mr Burns are eating the chocolate and see the photo, you can see that Marge's necklace changes from red to white and then back to red.
Lisa's Sax - S9-E3
Continuity mistake: In the episode 'Lisa's Sax' (season 9, ep. 3), King Toot's music store is across the street from Moe's Tavern. However, in numerous later episodes, there are stills of Moe's Tavern that show King Toot's music store next door, not across the street as in the aforementioned episode.

The Simpsons Christmas Special: Simpsons Roasting on an Open Fire - S1-E1
Continuity mistake: When Marge reveals the Christmas money from her hair, her necklace changes from red to white. (00:05:30)
Treehouse of Horror XIV - S15-E1
Continuity mistake: When the younger Professor Frink is receiving and wearing his Nobel Prize, his father goes nuts and starts stealing people's brains. The younger Frink figures out that kicking the elder Frink in the crotch will stop him. When he does, and the elder Frink falls off the stage, he is wearing the Nobel Prize, not the younger. (00:15:00)
Cape Feare - S5-E2
Continuity mistake: At the beginning of the episode when we see MacBain on TV he goes through a curtain, the curtain stays open for a second even though he isn't touching it.
Sunday, Cruddy Sunday - S10-E12
Factual error: At the end, when Homer and company are being carried out of the Super Bowl after the game, it is daylight. Usually, it is dark by the 4th quarter.
My Mother the Carjacker - S15-E2
Continuity mistake: At the end of the episode when Homer is looking for another message from his mom to find out if she's really alive, Marge and the kids enter the kitchen, Marge says that it's 2 in the morning. But the clock on the wall shows that it's 9 pm.
Mr. Lisa Goes to Washington - S3-E2
Audio problem: When Lisa is reading her speech for the essay contest, what she says does not match her mouth movements. (00:06:30)
Thirty Minutes Over Tokyo - S10-E23
Factual error: There is a flight to Honalulu, Hawaii listed on the national flights screen. This is a domestic flight, as it does not involve leaving the country.
Three Men and a Comic Book - S2-E21
Continuity mistake: When there is a rotating shot of Bart, Martin and Milhouse up in the treehouse, the shot never goes past the table, even though it should.
Homer's Barbershop Quartet - S5-E1
Other mistake: When the Be Sharps are singing at the Statue of Liberty, in some shots the stars in the sky are black.
E-I-E-I-(Annoyed Grunt) - S11-E5
Revealing mistake: After Lisa says "For a product that's evil and deadly" while standing in front of the tomacco stand, there's a duplicate Lisa behind her that stands still while she moves, giving Lisa the appearance of having two heads. It looks like the animators made a mistake while trying to animate Lisa.