Continuity mistake: Bart's guitar is lefthanded. So when the righthanded Otto plays it on the bus, the guitar is flipped over, with the little speaker on the top. But when Otto is playing it later in the garage, it is a normal righthanded guitar, with the little speaker on the bottom.
Continuity mistake: When Lisa is on the television talking about her tax rise, we see inside Moe's tavern. Look at where Lenny and Carl are sitting because they have moved 3 or 4 seats down when we go back to them later on. And there is no reason why they would they move like that.
Continuity mistake: In the cinema, it is clear that the people attending the movie keep on changing. For example, For a few shots, you can see that Krabappel and Skinner are sitting in The Front row, but the next time you see those seats, they have been replaced by Lenny and Carl and Krabappel and Skinner aren't in sight.
Continuity mistake: When Homer is stuck in the hole after saving the plant, Marge is seen in the background wearing lipstick. When Mr. Burns asks if there is anything he can do for them, there is a closeup of Marge and her lipstick is gone. When they cut back to Homer, Marge is again wearing lipstick.
Continuity mistake: When Homer is threatening to boot the Prime Minister, he is holding him still with his left arm. When the shot goes to behind Homer, he has his right arm on the Prime Minister's shoulder. When the shot cuts back to a direct view of Homer, his right arm isn't touching the PM.