Continuity mistake: In the episode 'Lisa's Sax' (season 9, ep. 3), King Toot's music store is across the street from Moe's Tavern. However, in numerous later episodes, there are stills of Moe's Tavern that show King Toot's music store next door, not across the street as in the aforementioned episode.
Continuity mistake: When the younger Professor Frink is receiving and wearing his Nobel Prize, his father goes nuts and starts stealing people's brains. The younger Frink figures out that kicking the elder Frink in the crotch will stop him. When he does, and the elder Frink falls off the stage, he is wearing the Nobel Prize, not the younger.