Deliberate mistake: When Marge goes offroading to get home there is a grassy field which leads right up into the Simpson driveway. However, in all other episodes there is no grassy field and instead there is a house in the spot where the field would be.
Continuity mistake: Springy the Olympic spring that Homer makes has 9 coils. Then we see the large version that has many coils, at the Olympics meeting. Homer's original spring is in his hand and now it has many coils also. When it cuts back to the stage, the large spring has few coils, like the very first time we saw Springy.
Revealing mistake: At court, as Homer is walking towards Alec Baldwin, Kim Basinger and Ron Howard, Bart is in the background sitting with the rest of the Simpsons, but his head is missing from his body. His head then reappears back on his body after about 2 seconds.