Continuity mistake: In the scene where Homer decides he wants to stay with Lurlene, she locks the door about 3 times before going to kiss Homer, but when Homer leaves, he just opens the door, he doesn't unlock anything.
Continuity mistake: The "dummy" in the episode changes consistently from a real character to a dummy sometimes. When Mel jumps into the vehicle in the museum the "dummy" moves aside. When Mel shoves it out of the way, it stays in one rigid position as if it is a dummy. During the chase, the "dummy" falls on the car behind and its facial expression changes. But when one of the guys from THAT car (not the one Mel was in) goes to the dummy, its head is ripped off and you can see that it is stuffed.

Tree House of Horror XII - S13-E1
Continuity mistake: In 'Hex and the City,' as they walk through Ethnictown, Marge has a handbag over her shoulder. But in the closeup when she says "Oh, listen.", the bag has gone. (00:01:15)

Continuity mistake: When Krusty the Clown arrives at the beauty pageant his tie is done up. When he goes on stage it is undone.

Continuity mistake: When the Repo Man takes Marge's ring, we see the ring on her left hand. It wasn't visible in the shot before. (00:14:40)
Lisa's Pony - S3-E8
Continuity mistake: When Homer tells the family where the horse will be kept, if you look in the background there are drapes on the wall. These drapes weren't visible when Marge and Homer walked in the kitchen.
Continuity mistake: When Bart arrives home after his bad day, he is seen wearing his lucky red hat. It was never visible anywhere before he got home. (00:07:10)
The Simpsons Halloween Special: Tree House of Horror - S2-E3
Continuity mistake: After the aliens have dropped off the Simpsons in their back yard, Marge is seen wearing her usual red pearls. Except in the final shot of the family. Then her pearls are white. This is not due to the alien light source altering the colors in the segment, because Marge's pearls have been a dark color through the whole episode, even inside the alien spaceship.
Skinner's Sense of Snow - S12-E8
Continuity mistake: When the children are sleeping in the canteen at school, Bart is resting his head on a red sweater. In the close up of him and Lisa, the sweater has now turned dark green. It's not because of the dark that we see it like that because there is a bit of light at the beginning of the next shot and we see the jumper to be dark green.
Continuity mistake: The "Stinger" only appears on the boxing gloves after Moe passes them to Homer. It is not there in all preceding shots.
Sunday, Cruddy Sunday - S10-E12
Continuity mistake: When Homer tells Lisa who is going with him to the Super Bowl, he says Comic Book Guy is going. In the shot showing the people lining up, Comic Book Guy is not there. He then appears on the bus later on.
Tree House of Horror X - S11-E4
Continuity mistake: When The Simpsons go home they find the words 'I know what You Did' on their door, written in blood. Then another shot after, you can see the door with nothing on it.
Continuity mistake: After Homer pushes in front of Lisa in the queue for the bathroom, she loses the toothbrush she was holding in her left hand. (00:01:15)

Tree House of Horror III: The Simpson's Halloween Special III - S4-E5
Continuity mistake: When Homer is escorting his family out of the house to escape the Zombies, he leaves the door wide open, but in the very next shot it is shut. (00:18:50)
Springfield Connection - S6-E23
Continuity mistake: When Marge is reading the books on death sports the police arrive and she turns around, you can see the two posts with the round balls on top with no words on it. But when Marge starts walking towards the police station it says "Police" on the balls and the post changes size. Also the door to the police station changes from brown to blue to clear glass and back to blue during the scene.

Continuity mistake: While Bart starts the I.Q. quiz and is reading it out loud, his teacher says "Shh. Visualize it Bart." While this happens if you look in the background you can see the alphabet line. It starts out Tt Uu Vv Ww, then Bart's teacher blocks out some, then you see the letter T again. (00:04:50)

Life on the Fast Lane (a.k.a. Jacques to Be Wild) - S1-E9
Continuity mistake: At work, the salad in Homer's sandwich disappears when he mentions that 'she' made it.
Continuity mistake: When Bart says "Everybody, come quick!", the fridge door goes from open to closed.
The Last Temptation of Homer - S5-E9
Continuity mistake: When Bart has had his throat sprayed, we see him standing up just before the shot changes to the power plant. The shine on his shoes instantly disappears and reappears several times.

Continuity mistake: After Homer dodges the cannonball near the end of the episode, the blue platform that he stood on goes further away from the edge of the stage between shots, and when Homer hugs Marge the platform has disappeared and Peter Frampton's cooler has become closer to Homer.