
Other mistake: How can Sideshow Bob make the "I H8 BART" licence plate if there are no number keys on the keyboard? (00:04:45)
OL1V3R666
Continuity mistake: When Homer and Marge come home from the dinner, Marge is wearing a fancy dress. But when Bart falls out of the house right after Homer walks in, she has her typical green dress on.

Continuity mistake: When Homer opens up his bar there are 4 cars parked outside, but later when Moe opens the garage door there are no cars, but no one has left.

Home Sweet Home-Dum-Diddly Doodly - S7-E3
Continuity mistake: At the Flanders' house, in the kitchen, Bart and Lisa have sandwiches on plates in front of them. The shot changes and everyone on the table has an empty bowl including Bart and Lisa.
David Mercier
$pringfield (or How I Learned to Stop Worrying and Love Legalized Gambling) - S5-E10
Continuity mistake: As Bart is on the phone trying to book his new act, you can see there is a split down his white shirt. When the shot changes and he tells Lisa to be quiet you can see that split has disappeared. (00:15:05)
Lummie

Continuity mistake: When Homer apologizes to Moe at the end, he comes in while the door is left open. It remains closed for the rest of the scene. (00:21:05)
Yoshi
Moaning Lisa - S1-E6
Continuity mistake: When Marge says, "Oh Homer, you'll lose your head if it wasn't secured to your neck," her necklace is white. However, when she gives the cup cakes to Homer and Bart, it is red. (00:02:00)

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. (00:02:40 - 00:07:05)
OL1V3R666
Burns Verkaufen der Kraftwerk - S3-E11
Continuity mistake: When Lenny tells Homer he's the safety inspector, the donut and paper cup have disappeared from his hands. (00:10:05)
Yoshi
Tree House of Horror XII - S13-E1
Continuity mistake: In 'House of Whacks,' Homer swings an axe at the water softener, making three large dents. The shot changes and one of the dents near the bottom vanishes. (00:13:30)
Hamster

Continuity mistake: When the family are driving home from the cinema, Bart draws an angry face on the back of Homer's head. It is never there at any other point in that night.

Continuity mistake: At the beginning, if you watch the clock as Homer walks in, the time changes between shots. (00:01:30)
Yoshi
Tree House of Horror V - S6-E6
Continuity mistake: At the end of 'The Shining, ' the family sits in the snow, Homer holding Willie's portable TV. Later, when they're frozen, Homer is holding nothing, although the TV's snow-topped antenna is still there. Additionally, Homer's left shoulder is missing. This is all due to the lack of correct colorization.

White Christmas Blues - S25-E8
Continuity mistake: While Mayor Quimby is giving his speech, Moe is in two places at once. He is simultaneously standing next to Mayor Quimby and amongst the crowd of people.
Nikolai
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.
Not true, watching a clip on Youtube and the episode on Disney+ the scratch is present. So it's not the device being used by the submitter.
Ssiscool

Continuity mistake: The card Bart has written for Milhouse is blank inside. However when Lisa looks at the card, the pun of the joke (on The Front of the card) is now there. (00:01:15)
Ssiscool

Character mistake: When Bart fakes being sick, on his way to the infirmary he passes the door with a sign that reads "maintanence" but the proper spelling is "maintenance", and when Bart reaches the infirmary, the sign on the door reads "ineirmary." (00:07:30)
Super Grover

Continuity mistake: Before Krusty is arrested, he grabs a drink and there is no olive in it. When he sits down, an olive has appeared in the drink. (00:05:35)