Continuity mistake: When the door is answered to Mr Burns and Mr Smithers, Barney is seen answering, who has orange sleeves. However when the door is opened, you can just make out a white sleeve, which could be none other than Homer's.
Revealing mistake: When Bart is swinging on the clothes line into the night, it is obvious that the brightness has been turned down because the sky is the same colour as in the previous shot. If it were night time, then the sky would be a dark purple colour.
Continuity mistake: In the cafeteria scene where Milhouse shoots a spitball at Kearney, in the shot when Milhouse says, "We have order, but at what price," Milhouse, Lewis and the bullies all change positions at their tables. You may have to watch this part several times to understand it.
Plot hole: When Marge is looking for inspiration for her novel, she looks at the boat painting in the TV room. On the frame, she reads it and says: "Scene from Moby Dick". Yet in "The Trouble with Trillions", Marge said that she painted it herself for Homer. So then why would she have to read the label to find out where it's from if she painted it?