Continuity mistake: When the doctor brings the "Hair in a Drum" product from under his desk, you can see in front of Homer there is a bottle of Dimoxinil and two gravity boots. When we see a close of the product in the next shot, the bottle and gravity boots on the desk have disappeared.
Other mistake: When the town is singing the Christmas carol you can see Sideshow Bob, who was arrested and jailed in series 1. We know he hasn't been released, as that's a plot point over the show. (00:18:10)Ssiscool
Continuity mistake: When Mr Burns tells Homer to wreck something, Homer throws a plant pot onto the floor and it breaks into several pieces. When Mr Burns tells Smithers to take him home, the pieces of the broken plant pot are nowhere to be seen.
Continuity mistake: In the meeting with Mr Burns and his advisers, one of the advisers holds up a photo of a guy who dated Mary Bailey and the adviser's tie is green. When he is sitting down in the next shot, his tie is now yellow.
Continuity mistake: When Mary Bailey has a group of interviewers around her her tie is blue, but in a close up of Mary in the next shot, her tie is purple.