Continuity mistake: After the safety expedition has taken place, we see that when everyone is in Burns' office that by his main doors there are 4 buttons on the wall. Yet when a shot from further away is shown, they are not there.
Continuity mistake: When the flashback of Ned and Todd Flanders singing in the Simpsons' doorway, Homer says he could use a TV tray. Then it goes back to the present time and Homer picks up a beer from the TV tray which is next to the couch. The TV tray was never there at any other point in the show.
Continuity mistake: The "dummy" in the episode changes consistently from a real character to a dummy sometimes. When Mel jumps into the vehicle in the museum the "dummy" moves aside. When Mel shoves it out of the way, it stays in one rigid position as if it is a dummy. During the chase, the "dummy" falls on the car behind and its facial expression changes. But when one of the guys from THAT car (not the one Mel was in) goes to the dummy, its head is ripped off and you can see that it is stuffed.