Goof (incorrectly regarded as a mistake): ... almost everything. This is a comedy animation series, and while efforts are certainly made to create a vaguely consistent setting in which mostly consistent characters live and work, the emphasis is clearly on laughs. Individual episodes must be consistent within themselves, but rigid continuity between episodes is unnecessary. Indeed, additional humor has been wrung, during the life of the show, from the very fact that the episodes are inconsistent with each other. No further "In one episode they say... but then in another episode they say..." goofs can be accepted.
Goof (plot holes): In one episode we learn that the octuplets are older than Maggie yet Maggie was in the show even before Apu get married.
Goof (continuity error): Marge is giving Abe a haircut. She walks off camera with the scissors in her hand but in the next shot, the scissors are on the table.
Goof (errors made by characters, possibly deliberate errors by the filmmakers): In a few episodes, the twins, who are supposed to be white colored, are yellow in brief scenes.
Goof (continuity error): Springfield Elementary School is yellow in some episodes and white in others, sometimes changing color multiple times in the same episode (later episodes corrected this continuity problem by making the school's official color yellow, and including a picture of the "old, white" school in Principal Skinner's office.)
Goof (continuity error): When Moe appears in the first season he has black hair. In all subsequent seasons his hair is a grayish-blue.
Goof (continuity error): Some characters have brown skin in some episodes and yellow skin in others. Two examples would be Hans Moleman and Judge Snyder.
Goof (continuity error): Marge's father in The Way We Were looks nothing like he does in Fear Of Flying (although it's possible the latter is her real father and the former is her stepfather.)
Goof (continuity error): Ralph is in Bart's class in some episodes and Lisa's in others.
Goof (continuity error): Although the giant stone head in the Simpson basement is gray, in some episodes it has appeared rainbow-colored.
Goof (continuity error): In "The Itchy And Scratchy And Poochie Show" episode, in the comic store at the autograph signing, the Rainer Wolfcastle poster in the background spells his character's name as "McBane." In all other episodes it's spelled "McBain."
Goof (continuity error): Roger Meyers Jr. fires a worker in the conference room in "The Itchy And Scratchy And Poochie Show." That worker is gone in all subsequent scenes but was never shown leaving the conference room.
Goof (continuity error): In "Two Cars In Every Garage And Three Eyes On Every Fish", Mr. Burns tries to bribe the nuclear inspector by stuffing wads of cash in his pocket. When the inspector leaves the wads of bills in his pockets are gone.
Goof (continuity error): When the older, male judge character is introduces in "Bart Gets Hit By A Car", he is referred to as Judge Moulton. In all subsequent episodes/reference material/etc., he is called Judge Snyder.
Goof (revealing mistake): When Homer comes to terms with himself after stealing cable and looks out the barred windows, the house guests behind him are drawn without facial features.
Goof (revealing mistake): Jasper confesses to being blind when viewing the premier of the Poochie episode, but could see perfectly well in a number of episodes (i.e. watching the Tatum fight at the Simpson house and walking down the street when Smithers shot him in his wooden leg.)
Goof (continuity error): When wearing her normal green dress, Marge wears a red necklace. However, often she is shown for a single frame or two wearing a white one (generally this happens in close-up shots of her head.) This happens in numerous episodes.
Goof (continuity error): Louis, Bart's black friend, appears yellow in one frame in "The Crepes Of Wrath" when Principal Skinner is introducing his mother to Bart and his friends.
Goof (continuity error): In Bart's Comet, Apu has yellow skin in one frame.
Goof (continuity error): Chief Wiggum's proper hair color is blue. A common animation error is for it to be black in certain scenes (this generally happens in night scenes, or other scenes in dark places.)
Goof (errors made by characters, possibly deliberate errors by the filmmakers): Grandpa's World War II squadron is usually called the "Flying Hellfish", but in some episodes the stationery and memorial read "Fighting Hellfish."
Goof (continuity error): Fat Tony's real name changes from episode to episode, though possibly intentionally because he might have several mafia names that the police know of.
Goof (continuity error): The family portraits in the Hibbert home have yellow skin.
Goof (continuity error): When Homer visits the Twin Towers and is forced to scale both towers in search of a working bathroom, for one frame neither tower has the radio antenna on top.
Goof (continuity error): When Milhouse Van Houten appears in the pilot episode, he has black hair. In all subsequent episodes, he has blue hair.
Goof (continuity error): The interior of the Simpson's house changes from time to time, e.g. the size of the archway and what room it leads to.
Goof (incorrectly regarded as a mistake): When Waylon Smithers makes his first appearance in "Homer's Odyssey", he has orange skin. He has the standard yellow flesh in all subsequent appearances. This was actually a change that the creators made knowingly. Smithers was mistakenly animated with the wrong color and was made an African-American by Gyorgi Peluci, the color stylist. David Silverman has claimed that Smithers was always intended to be "Mr Burns' white sycophant," and the staff thought it "would be a bad idea to have a black sub-servient character" and so switched him to his intended color for his next episode, the fourth episode aired.
Goof (continuity error): In the intro sequence, Lisa is playing a blue sax, but during the actual show she has a golden sax (this was corrected in the new HD intro).
Goof (continuity error): When Millhouse, Bart and Lewis are talking about the fake Santa, Millhouse's eyebrows go from blue to black to blue again.
Goof (continuity error): When Marge tells Homer to close his eyes as she takes out the money jar, her necklace is red. When she takes it out of her hair in the next shot, the necklace is white.