Continuity mistake: When Homer is preparing the barbecue, he is wearing an apron that says "Mafia staff apron". After he goes to pick up the lighter fluid, it reads "Maffia staff apron", and after he has lit the barbecue, it is back to saying "Mafia".
Other mistake: When Homer is at Moe's, there is a boxing match happening on TV. There is a bald guy who is getting considerably battered by the opponent and gets knocked out, and then for some absurd reason he is victoriously standing as if he had won.
Continuity mistake: John is wearing Homer's old "Pin-pals" bowling shirt when the Simpsons first meet him, but only after we see the back of the shirt does the name "Homer" appear over John's breast-pocket; clearly there was nothing there during their entire previous conversation. And even if this is a deliberate "mistake" for comic purposes, in 7th season episode"Team Homer", the very same bowling shirt is ripped to shreds when Homer is being mauled by Mr.Burns' dogs at the very end.
Revealing mistake: In the scene where Marge's coat pocket bursts open, the candy flows onto the floor. However if you look to the right of her some of the candy seems to almost float above the floor. The candy's direction is completely to the right but because of the floor being angled it spreads over one woman who is standing and there is nothing there to support the candy.
Other mistake: When the Simpsons' drives the electaurus off the edge of the harbour, it is shown that the front of the car goes downwards. When it cuts to the interior view with the Simpsons' screaming, the view from the back window is tilting down, suggesting the rear is going down and the front coming up. But in the next shot the car is seen in the water with the front down.