Continuity mistake: Bart yanks Homer's beard off in the mall, Homer then takes Bart into Santa's workshop. When they are inside Santa's workshop, Homer's beard is back on, despite the fact that he didn't put it back on, he let it fall to the ground when Bart pulled it off.
Continuity mistake: In the bit where they sit on the sofa at the start of each episode, there is one rather long one where it starts off with the Simpsons dancing in a line, then gradually evolves into a circus with elephants and men dancing around fire. Look closely in between Homer's right leg and the dancing woman on his right. There is a pile of books on the little shelf underneath the lamp and telephone, and just as the camera starts zooming out from the line of dancing, the orange book on the left noticeably changes to purple.
Continuity mistake: Marge drops Lisa off at school for band practice, and she walks up to the entrance steps, and then converses with two boys. Then her teacher comes up, and Marge gets annoyed. When she comes to grab Lisa back into the car, Lisa is right by the roadside, along with the two boys and the music teacher.
Audio problem: In the episode where Homer is hosting gay marriages, at one point Homer is strangling Bart and tells Lisa to hold his spot. You can hear Lisa say, "Why you little," and you can hear Bart choking, but neither of their mouths is moving.
Continuity mistake: Just before the kitchen mistake, Marge walks through the doorway from a room with a piano, beside a bare wall, and on the piano are two frames and music sheets. But when Marge turns the jar upside down, there is a large picture on the wall beside the piano, and the music sheets and frames have disappeared from the piano.
Continuity mistake: When Mr. Burns comes on the P.A. system Homer started eating a donut, he takes a bite and after that the donut stays the same size even though he takes 2 more bites
Continuity mistake: In "Time and Punishment", when Homer returns to the basement after his first time travel, there is a scorch mark on the work bench where the toaster was placed when he turned it on. The angle changes to a shot from the stairs, and the scorch mark is gone. Later in the episode, the mark is back on the bench.