Continuity mistake: Level 9 "Invasion of the Yokel-Snatchers" cutscene: When Cletus tells the aliens that he spent all day clipping coupons for a "flat screen magic picture machine," we see the building behind him has nothing on its roof, but in the next shot a huge billboard has suddenly appeared on that building's roof.
Continuity mistake: Level 12 "Grand Theft Scratchy" cutscene: After Senator Clinton congratulates Marge and Lisa for their great work a large sign falls on her, and when the bag appears with the keycard inside, in the closeup the bag's strap is lying at the back of the bag under its flap, but in the next shot that strap is now lying on the opposite side of the bag, at the front, and the bag's position on the sidewalk changes as well.
Continuity mistake: Level 16 "Game Over" cutscene: When Bart grabs the save games and threatens to destroy it, we see the front labeled "Memo: Planet Earth..." and the back is blank, then when Bart holds it up the front is facing forward, but in the next shot the label faces the opposite way, note Bart's thumb.