Continuity mistake: After the dinner with Sideshow Bob, Patty, Selma and the Simpsons, the food on Bart's plate changes between shots. We first see a piece of meat with some peas, then we see two pieces of meat and some mashed potato. It changes randomly throughout the scene.
Continuity mistake: After Krusty is done doing the special promotion commercial, he is given a telegram informing him of the Soviet boycott. However, this is a flashback, and at this point in time Krusty is illiterate (as will later be established in season 1) so he shouldn't be able to read the telegram.
Continuity mistake: In the scene where Miss Hoover asks Lisa what the ninteenth-century figure nicknamed 'Old Hickory' is, there is no text on the covers of the book she's holding. In the next two shots, however, 'Teacher's Edition' can be seen on the front cover and the spine of the book.
Continuity mistake: When Homer is first interviewed for 'Rock Bottom', the clock behind him stays at 5 O'Clock for the duration of the interview, but when the interview is actually shown on the TV, the times keep changing. The hands on the clock are also slightly different.