Continuity mistake: In the scene after Homer and Apu rescues the babies, Maggie is first seen wearing a bow. Then, when she's next seen, it's gone. When we see her one more time, the bow is back.
Continuity mistake: In the initial robbery at the Kwik-E-Mart, Apu doesn't realise what's going on until the fake Krusty demands the money from him. Yet in the video shown in court, he turns to face the fake Krusty right as Homer screams.
DaveJBContinuity mistake: Lisa places her Gaspacho on the house end of the table but in a following shot it's at the other end when the dog is drinking it.
Continuity mistake: When Homer is at Moe's tavern talking about Marge, the level in his beer glass keeps changing even if he doesn't drink.
Dr WilsonContinuity mistake: When the person delivers the animation cell, his skin changes from white, to brown and then back to white. When Bart looks through the letterbox to see who was there, his skin is white. When he gives the box to Bart, it is brown. When he hits Bart in the face and says "that's for keeping me waiting", it is white again.
Continuity mistake: At the bar, we see Mrs. Krabappel, and she disappears for the rest of the scene. (00:13:40)
YoshiSuggested correction: The place is crowded, she can easily be anywhere in the packed bar and simply out of sight.
Ssiscool
Continuity mistake: When Skinner is set free, while holding onto the vacuum, he changes position between shots. (00:20:15)
YoshiSuggested correction: In the first shot he is holding on using on his one free arm - as soon as his second arm is free he grabs on with that as well, this is the change of position listed.
Ssiscool 
Continuity mistake: In the beginning, when Bart finds that the dog ate his homework, the poster on Bart's door has disappeared for one shot. (00:01:45)
Yoshi
Continuity mistake: When the mobsters enter Skinner's office, the phone on his desk changes position between shots. Also, the name bar on his desk doesn't read anything on it. (00:13:50)
YoshiThree Men and a Comic Book - S2-E21
Continuity mistake: When Bart, Martin and Milhouse start to fight, the Radioactiveman issue 1 that they were protecting has disappeared from the table. Then when Marge brings up some food, they all start eating. The table has disappeared altogether.
The Simpsons Christmas Special: Simpsons Roasting on an Open Fire - S1-E1
Continuity mistake: When Homer and Flanders drop their presents, the presents move around and change colours between shots. (00:10:20)
Hamster Continuity mistake: John is wearing Homer's old "Pin-pals" bowling shirt when the Simpsons first meet him, but only after we see the back of the shirt does the name "Homer" appear over John's breast-pocket; clearly there was nothing there during their entire previous conversation. And even if this is a deliberate "mistake" for comic purposes, in 7th season episode"Team Homer", the very same bowling shirt is ripped to shreds when Homer is being mauled by Mr.Burns' dogs at the very end.
Homer vs. Lisa and the 8th Commandment - S2-E13
Continuity mistake: When Homer and Marge are discussing about unhooking cable, the time on the clock on Homer's bedside table changes between shots.
OL1V3R666Homer vs. Lisa and the 8th Commandment - S2-E13
Continuity mistake: When all the kids are watching TV, we see Wendell, Martin and Milhouse. Absurdly they appear to be right below the ceiling in this shot - when in every other shot they are sitting on the sofa or standing.
OL1V3R666The Last Temptation of Homer - S5-E9
Continuity mistake: When Homer and Barney are at Moe's, Barney drinks his beer and then his cup his empty, and Homer's is full. Then later, Homer's cup is empty and Barney's cup has magically refilled itself.
OL1V3R666I'm Goin' to Praiseland - S12-E19
Continuity mistake: When Flanders' girlfriend pulls up at the end, from one angle she's wearing a pink shirt with what looks like a white vest over top. But from another angle her shirt is all white.
Krista
Tree House of Horror VI - S7-E6
Continuity mistake: In the 'Nightmare on Evergreen Terrace' segment, in Bart's dream you see a shadow of what looks like someone with four knife-fingers. Then when we see who it is, it is Willie holding a rake. But the rake has five tines, not four, so the shadow is wrong. (00:07:00)
Hamster Continuity mistake: When Marge arrives after the song, Flanders and Lovejoy are talking to her, and behind them you can see a crowd of people. The crowd then disappears in one shot, only to reappear a shot later.
Hamster Homer Badman - S6-E9
Continuity mistake: Homers car gets blown over by a helicopter, and gets dented, and all the windows smash. But in the next overhead shot the car is parked on the side of the road, perfectly intact.
Days of Wine and D'Oh'ses - S11-E18
Continuity mistake: When Moe tells Homer to sit on Barney's seat in front of the beer tap, Homer is sitting on a stool next to it. Homer suddenly switches seats between shots and the other seat disappears.
