Continuity mistake: During the song "Its the American Way" Snake is robbing Hans Moleman during Chief Wiggum singing and Snake's hair is blue. His hair has always been a brown-red color. In the very next scene to confirm its a mistake you see him in the park saying "Yo Sherry Bobbins" and his hair is back to its brown-red color.
Continuity mistake: When the family are watching Krusty Komedy Klassics, at the beginning of the show, they all have grey tables, with their dinner. But when Bart asks for milk, Marge gets up to get it and her table has disappeared. When she comes back the table has re-appeared.
Deliberate mistake: During the final musical number, just after Marge sings "I'm going bald from stress," the family sings "But we're happy just the way we are," but Homer's mouth doesn't move, because the producers forgot to record Dan Castellaneta's lines.