Continuity mistake: When Homer is watching TV at the beginning of the episode, Marge takes away his plate of leftovers and puts an ice-cream sundae in front of him on his tray. When we see him a few shots later, his plate is now back on the tray with the ice-cream sundae on top of it.
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)
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.