Brother From Another Series - S8-E16
Continuity mistake: Sideshow Bob and Cecil's glasses keep switching from wine glasses to champagne glasses while they are having dinner.
Lisa vs. Malibu Stacy - S5-E14
Revealing mistake: When Lisa says "Now all we need is a name", Maggie's head disappears for a split second right before the shot changes to Bart talking.
Lisa's Pony - S3-E8
Other mistake: When Lisa rings Homer to tell him that she loves him, the shot of her talking is re-used from "Bart's Dog Gets an F." We can tell this because her cheeks inexplicably appear puffed up (in "Bart's Dog Gets an F" she had mumps).
When You Dish Upon a Star - S10-E5
Revealing mistake: At court, as Homer is walking towards Alec Baldwin, Kim Basinger and Ron Howard, Bart is in the background sitting with the rest of the Simpsons, but his head is missing from his body. His head then reappears back on his body after about 2 seconds.
Treehouse of Horror XVIII - S19-E5
Factual error: In the Mr. And Mrs. Simpson segment Homer is firing a minigun at Marge and three of the barrels are seen spitting fire. A minigun however only uses one barrel at a time to fire.
Revealing mistake: When Homer enters the Kwik-E-Mart, the automatic door doesn't fully open, and Homer's left arm goes right through it. (00:03:40)
Lisa's Date with Density - S8-E7
Continuity mistake: In Lisa's music class, there is a wide shot of the entire class. Üter sits next to Milhouse Van Houten in this shot. However, in the next shot, it's Ralph Wiggum. In a following shot of the entire class, it's Üter again. They both replace each other in the same seat.
Character mistake: Moe says that no women have been to his bar since 1979, yet in "Flaming Moe's" he hires a female waitress. Also, in the episodes "New Kid on the Block" and "Homer the Vigilante," Ruth Powers appears in the bar, and orders a beer in the former episode while Marge visited the bar in "Bart Gets Hit By a Car."
Revealing mistake: When everyone is cheering for Ned at the neighborhood watch meeting, Apu's hand is yellow.
Revealing mistake: When Homer is on the news talk-show talking to the cat burglar, he puts his hand on his chest and the opening of his shirt appears over his hand.
Revealing mistake: When Homer pushes the soda vending machine next to the candy vending machine, a large scratch appears near the upper right corner of the screen.
Suggested correction: This is in the questions as well. With the long amount of time the artists have to make an episode it's very hard that something like this would get by them. It's probably the device you watched it on.
One Fish, Two Fish, Blowfish, Blue Fish - S2-E11
Revealing mistake: In the credits it shows Homer eating pork rinds. This is a reused scene from "Moaning Lisa."
Audio problem: During the party, when Princess Kashmir asks Homer to dance, her mouth doesn't move. (00:08:10)
Two Cars in Every Garage and Three Eyes on Every Fish - S2-E4
Continuity mistake: When Mr. Burns is drinking on his desk, a frame is misplaced. His shot glass, for a split-second, moves from its original position, to the edge of the table, and back.
Ned 'N' Edna's Blend - S23-E21
Continuity mistake: When Homer (playing Jesus) is on the cross and speaking to Snake, his stubble briefly flickers to yellow twice in the same shot.
Other mistake: In this episode, Smithers' skin is brown, however in future episodes, his skin is yellow. (00:05:00)
Sideshow Bob's Last Gleaming - S7-E9
Factual error: Sideshow Bob couldn't hide out in the Duff blimp's helium pocket because there wouldn't be any oxygen.
Brother's Little Helper - S11-E2
Continuity mistake: When Homer and Marge are in the research lab, the scientist pulls off a white cloth which is covering the hamster tank and places a tube next to it, but in the next shot it's gone. (00:04:55)
The Bart Wants What It Wants - S13-E11
Other mistake: Ned Flanders wins the $100 bill, but on the side it says $50.