Lisa vs. Malibu Stacy - S5-E14
Revealing mistake: When Lisa says "Now all we need is a name", Maggie's head disappears for a split second right before the shot changes to Bart talking.
Revealing mistake: When everyone is cheering for Ned at the neighborhood watch meeting, Apu's hand is yellow.
Revealing mistake: When Homer is on the news talk-show talking to the cat burglar, he puts his hand on his chest and the opening of his shirt appears over his hand.
Revealing mistake: When Homer pushes the soda vending machine next to the candy vending machine, a large scratch appears near the upper right corner of the screen.
Suggested correction: This is in the questions as well. With the long amount of time the artists have to make an episode it's very hard that something like this would get by them. It's probably the device you watched it on.
Continuity mistake: When Homer and Marge are in bed Marge moves up and we can see she isn't wearing anything, but in the close up you can see that she is wearing her pyjamas - note the thing around her neck, then it's back to nothing.
Cape Feare - S5-E2
Continuity mistake: When Bob has finished singing the score of the H.M.S. Pinafore, an English flag comes down. But when he draws his sword to kill Bart, the flag is gone. Where could it have gone?
Continuity mistake: When Barney becomes his old self again at the training center, he gets on a jetpack while holding a bottle. In the next shot while he is in the air, the bottle is gone.
$pringfield (or How I Learned to Stop Worrying and Love Legalized Gambling) - S5-E10
Continuity mistake: When Marge enters the bedroom, the family appears from behind the mattress, and Maggie's outfit is her traditional blue, but when Marge picks her up her outfit is white. (00:17:10)
Continuity mistake: Apu grabs the ham and crosses out of the expiration date with a thick black marker. When he places the ham on the shelf the black marker cross out is gone and the writing on the pack looks perfectly printed as before.
Revealing mistake: When Moe grabs Ned Flanders' shirt at the bar, Ned's torso is incorrectly drawn over his arms as well as the mug he is holding. (00:10:50)
Continuity mistake: At the "Do What You Feel" Festival, Homer shows up in his house robe, bear feet styled slippers, unshaven, and unbrushed hair. When he takes the float to save Bart, he's in his regular clothing of blue pants, white shirt, shoes, and clean shaven.
Continuity mistake: The cake that Marge bakes for Homer is initially white, but pink icing instantly appears on top between shots.
Audio problem: After everyone congratulates Ned, he goes to Homer and his mouth moves but nothing is said, and the first line afterwards is not in sync with his mouth movements.
Bart Gets an Elephant - S5-E17
Continuity mistake: Bart's elephant crushes the dog house, then it is shown standing, and then it is crushed again.
Revealing mistake: The people waiting in line for the game are extremely crude drawings. Some of them aren't even completely drawn, e.g, some are missing arms, legs and other body parts.
Lisa vs. Malibu Stacy - S5-E14
Continuity mistake: When the Simpsons are leaving the house to go to the mall. The aerial shot shows that they are leaving in Marge's red station wagon, but the very next shot of them in the car shows that they are in Homer's pink sedan. The shot of the station wagon leaving is probably a re-used shot from a previous episode.
Lisa vs. Malibu Stacy - S5-E14
Continuity mistake: When the Simpsons are about to leave the house to go to the mall, they're all wearing jackets, but when they're in the car just a few seconds later, none of them have their jackets on.
Revealing mistake: Apu's moustache disappears for some seconds after having his name tag removed.
Homer's Barbershop Quartet - S5-E1
Continuity mistake: Homer and Marge's bedroom has photos of Bart and Lisa when they were 10 and 8 years old. However, the story takes place when Bart and Lisa were only 3 and 1.
Revealing mistake: In one angle when the Flanders are having dinner, Ned's eyeglasses have no temples.