Continuity mistake: When Skinner is calmly driving the bus on the way to school, they have obviously gone past a number of power lines (you can see that they have because you can see them in the background). However, when Skinner is trying to get onto the main road, the power lines have disappered from the background.
Revealing mistake: When Bart's dog is chewing on the TV remote, one of the shots on it is of "I Can't Believe They Invented It." with Troy McClure as the host. This is exactly the same shot from "Homer vs Lisa and the 8th Commandment". You can tell because Troy's voice is different. The lip sync is also out, revealing that the line "As an actor, my eyeballs need to look their whitest" is dubbed in.
Plot hole: In the car chase, Chief Wiggum says that one of the suspects (Marge) "is wearing a green dress, pearls, and a lot of blue hair." However when we see Marge from the police car all we can see is her hair, so how would he have known about the pearls and dress?
Continuity mistake: In the beginning after Maggie hit Homer on the head, it shows him laying on the couch with Bart & Lisa on the left. The next shot it shows the TV, in the next shot it shows Homer on the left sitting and Bart & Lisa on the right. It's impossible for Homer to get up and switch sides with Bart And Lisa in less than 5 seconds.