Audio problem: When Patty says "Marry me Patty?" while reading Skinner's message, her mouth doesn't move.
The D'oh-cial Network - S23-E11
Continuity mistake: At one point in court while everyone is laughing, Patty's hair is colored wrong. Her hair is a similar shade of color as Rod and Todd (who are sitting next to her) rather than her own bluish grey hair.
Marge vs. the Monorail - S4-E12
Revealing mistake: When the passengers board the Monorail, Homer begins pushing two buttons. The second button he pushes is just the counter. There is no button there. (00:15:40)
Brother Can You Spare Two Dimes? - S3-E24
Other mistake: On the DVD, when Homer is sitting on the Spinemelter, he says at one point "I've tried to fill it with family, religion, community service, but those were dead ends. This card is the answer." However the subtitles say "I've tried to fill it with family, religion, but those things were dead ends. This card is the answer", leaving "community service" out.
Suggested correction: It is very common to shorten the subtitles so viewers don't have to read much.
Character mistake: When Chief Wiggum shows up to arrest Nelson during the game at the end of the episode, Bart goes with Wiggum instead while pretending to be Nelson. Even a cop as incompetent as Chief Wiggum would not have mistaken Bart for Nelson since he'd had numerous dealings with both before this episode; Wiggum may not be particularly bright, but he knows who's who in Springfield.