Gaffes

Ralph Wiggum is in Bart's class in some episodes and Lisa's in others. The same can be said about a few other recurring students.

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.)

In a few episodes, the twins, who are supposed to be sheet white, are yellow in brief scenes.

The interior of the family's home changes from time to time, e.g. the size of the archway and what room it leads to.

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.)

In the leader, Homer carries a rod of radioactive material around emitting a green light, but Cherenkov radiation looks blue.

Fat Tony's real name changes from episode to episode, but this is probably intentional because it is common for Mafia hit men (in popular culture, and possibly in reality as well) to have many different identities.

"In one episode they say this, but then in another episode they say that, and in yet another episode they say the other." As this is an animated comedy series, the emphasis is clearly on laughs rather than complete verisimilitude. Efforts are certainly made to create a vaguely consistent setting in which mostly consistent characters live and work, and many episodes refer to each other, but rigid consistency of every single detail in all episodes is unnecessary. In many episodes, the fact that something is inconsistent is the express point of a gag. Our general rule is that each episode is expected to be consistent within itself, but intra-episode inconsistencies are not being listed. There can be exceptions for unusually noteworthy matters, e.g., inconsistencies repeated in multiple episodes (such as the hair and skin colors of secondary characters, and the layouts of the main landmarks), drastic changes to a character's nature (such as Ralphie's school status, Jasper's abilities, or Milhouse's hair color), or something with an interesting anecdote behind it (such as Smithers' skin color).

The location of the stairs going to the basement in the Simpson house changes frequently throughout the series. Sometimes the entrance is located in the hallway off the kitchen that goes to the garage, while in other episodes it is in the foyer by the main staircase.

In many episodes, the locations of prominent buildings in Springfield move around.

The family portraits in the Hibbert home have yellow skin.

Jasper acts as if blind in some episodes but can clearly see in others.

In the opening sequence, when Homer leaves work for the day, Mr. Burns is in the background flicking his left wrist, presumably to check that his watch is running on time. But in "When Flanders Failed (1991)", it's established that Mr. Burns is left-handed; lefties predominantly wear their watches on their right wrists, because it's easier to work the clasp with their dominant hand.