Other mistake: When Lisa checks Marge's resume, she reads it and turns it over. But there is no text on the back as there should be. (00:06:30)
Continuity mistake: Homer drives to Moe's Tavern with his normal shirt on. He has his green jacket on top of it in the next shot. (00:04:45)
Bart's Friend Falls in Love - S3-E23
Continuity mistake: At the beginning of the episode when Bart runs away from Homer, Homer starts rolling down the stairs. The banister disappears between shots. After Homer gets to the bottom the banister reappears. (00:01:40)
Continuity mistake: In the video store, Homer grabs the 'Football's Greatest Injuries' tape with his right hand. He was also holding Maggie with his right arm, so he should have dropped her. (00:06:00)
Itchy and Scratchy: The Movie - S4-E6
Audio problem: Bart's mouth doesn't move when he laughs after saying "Now's our chance to be bad."
Marge vs. the Monorail - S4-E12
Continuity mistake: At the beginning of the episode, Homer slides down a big pipe and smashes through one of the windows on the right side of the car. But when Homer crashes into the chestnut tree, it is fully repaired. (00:01:25)
It's a Mad, Mad, Mad, Mad Marge - S11-E21
Continuity mistake: When Homer is changing the oil in Marge's car, he is in his own driveway, but in the next, he is perpendicular to Flanders' house and then in Flanders' driveway.
Suggested correction: He is in his own driveway the whole time. There's never a shot of him in a different position and certainly never in Flanders' house. You ever see the brick wall from the Simpsons' house in the last shot.
Like Father, Like Klown - S3-E6
Continuity mistake: At the end of Krusty's story, Homer changes from being in his tuxedo to being in his normal clothing. (00:10:10)
Suggested correction: It should be noted that a considerable amount of time has passed while Krusty is telling his story. Note the candles in the screenshot for example, they have burnt out during the story. Its reasonable to assume that there would have been a chance for them to change during this time.
Continuity mistake: When Otto is playing the guitar with bart in the middle of the night, there are close-ups of Otto whenever he is singing. You can see that the wall behind him is plain. However, when there is a wide shot of otto and bart, the wall behind otto has two tables leaned against them. (00:15:50)
Tree House of Horror II: A Simpsons Halloween - S3-E7
Continuity mistake: When Homer purchases the monkey's paw, you can see open desert to the right of Homer. There was a building in its place a few shots earlier. (00:02:30 - 00:03:15)
Other mistake: When The Town is singing the Christmas carol you can see Sideshow Bob, who was arrested and jailed in series 1. We know he hasn't been released, as that's a plot point over the show. (00:18:10)
Suggested correction: Maybe he was allowed to participate due to good behavior. That's a possibility.
Good behavior or not, a prison wouldn't furlough an inmate just so they could participate in a snow day.
I'm not sure if it's been discussed here or listed as a mistake. Part of the problem with "The Simpsons" and the Sideshow Bob character is that time both moves forward and doesn't move. In "A Brother From Another Series," Krusty visits the prison and tells Bob he hasn't seen him in years because Bob had been in prison all those years for his crimes. Yet Bart and Lisa are still the same age and in the same grade. Bob is also let out of prison on a work-release program in the episode.
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: In the new opening sequence, Kearney and Dolph cut off Jebbidah's head. But in the pan of the city, the head is back.
Burns Verkaufen der Kraftwerk - S3-E11
Continuity mistake: When Homer is at the vending machine, in certain shots he is wearing a tie and in others he isn't. (00:02:20)
Character mistake: Burkina Faso is listed as being among the Southern Hemisphere locations Bart called when Homer is looking over the phone bill; since Burkina Faso is completely within the Northern Hemisphere, it would have made no sense for Bart to call there.
Suggested correction: It may not make sense, but Bart isn't exactly the sharpest tool in the shed. In this same episode, he looked at his globe and thought Rand McNally was a country and was convinced by Lisa that the citizens there wear hats on their feet and that hamburgers eat people. Additionally, he could have just decided to prank call someone there if for no other reason than he thought the name of the country was funny.
Entirely possible, but since Bart had a globe right there to refer to, even someone with his limited geographical knowledge would have been able to tell that Burkina Faso is in the Northern hemisphere, unlike the other locations he did call as depicted in the montage (Antarctica, Argentina, unspecified South American nation, etc).
Homer's Triple Bypass - S4-E11
Continuity mistake: When Homer begins praying, the room is dark and there is barely any light. When the nurse shushes him, suddenly it's a lot brighter. (00:14:20)
Homer's Triple Bypass - S4-E11
Continuity mistake: When Barney and Moe visit Homer, the vase of flowers disappears and then suddenly reappears at the end of the scene. (00:16:10)
Marge vs. the Monorail - S4-E12
Other mistake: When Homer crashes his car, he just disappears when it hits the tree. (00:01:50)
Marge vs. the Monorail - S4-E12
Continuity mistake: When Grandpa Simpson stands up in The Town hall, the position and amount of paintings on the wall change several times between shots. (00:06:10)
Marge vs. the Monorail - S4-E12
Other mistake: Lanley's desk in his trailer suddenly gets brighter even though none of the lights in the trailer were turned on. (00:12:50)