Continuity mistake: Steve's hair is soaking wet when Francine jumps in the pool, yet dry as a bone when Francine emerges from the pool moments later. (00:03:45)
Continuity mistake: When Francine brings lemonade on a tray for Steve and his friends, she assumes Snot is drowning. Francine then drops the tray and it lands on the ground before jumping in the pool after him. The tray disappears when she emerges from the pool with Snot. (00:03:45)
Continuity mistake: Stan's crocodile skin tie changes back to his ordinary black tie when he says "Prison rules" to himself.
Continuity mistake: The poop on the edges of Roger's robe that Stan wears disappears in later shots.
Continuity mistake: We see an old lady shoot a yellow arrow at Roger's golf bag. The shot changes and when we see the arrow again it is red. (00:09:35)
The Kidney Stays in the Picture - S7-E16
Plot hole: It is explicitly stated in this episode that Hayley was conceived in 1996. While normal for shows without character aging to change birth years to benefit the plot, this move was premature. Hayley's permanent age is stated as 18, and this episode implies that in 2012 she would be much younger than this. The whole plot centres around her being legally able to drink in a bar, so this is a pretty big mistake.
Why Can't We Be Friends? - S8-E5
Factual error: When the guys are on the Graviton, you can tell they're spinning clockwise, or to their left. That means everything should be pushed in the opposite direction, but one girl's hair is flowing to her left when it should be flowing to her right.
Adventures in Hayleysitting - S8-E6
Other mistake: When Hayley is chasing Steve and his friends in her car, after Steve gets on the bus, Hayley turns around a corner to follow it. When this happens, the license plate on Hayley's car has nothing written on it.
Blood Crieth Unto Heaven - S8-E10
Continuity mistake: When Stan and Francine are first in the kitchen, a record player is briefly seen to the right of the screen. However, when Stan returns to the kitchen after all his dad's items have been put in the bag there is no sign of the record player in the other room. (00:03:15 - 00:03:55)
Other mistake: When Francine gives the family gifts, Stan writes "How did you pay for this?" on his board. Francine replies "Well, money, of course" to which Stan writes another sentence on his board. He starts writing without clearing the board, yet when he turns the board back to show the sentence "Is that true, Steve?", the original "How did you pay for this?" isn't visible on the board, despite not clearing it. (00:12:00)
Continuity mistake: Stan hears Francine's parents at the door, and Klaus is in front of him with a bowl of nachos to the left of Klaus. In a few shots, Klaus's bowl and the nachos switch spots.
Continuity mistake: Steve receives a Rowflux and five containers of protein powder. He takes one of the containers leaving only four still on the floor. In a subsequent scene, Steve is still holding the container but now five containers can be seen behind him, giving him six containers altogether.
Cock of the Sleepwalk - S9-E11
Continuity mistake: After Stan says "Fired?!" and the camera goes back to Bullock, the word IHOP appears on the board behind them. (00:13:45)
Character mistake: When Francine calls Stan to tell him Hayley is in rehab, he later asks her whose car she crashed, when Francine had never said that a car had been crashed to begin with.
The Devil Wears a Lapel Pin - S11-E7
Other mistake: Francine tells Roger that she's jealous that he's starting college, but in the pilot, she mentions that she met Stan in college.
Criss-Cross Applesauce: The Ballad of Billy Jesusworth - S11-E17
Continuity mistake: When the game is tied 10-10, Yao Ming and Roger are in the circle at half-court, but when Ming checks the ball back, they are out and in front of the circle and not at half-court anymore. When Roger starts to dribble, he is back at the half-court line, all without them moving.
Continuity mistake: When Thacker is killed, blood splatters on his campaign posters resting against the couch. When security breaks in and shoots Sturges, the blood on the posters is gone.
Factual error: Klaus shoots 25 rounds out of a pistol. Given the size of the pistol, that many rounds would require an extended magazine, but it's a normal magazine that isn't extended.
The Life and Times of Stan Smith - S12-E15
Character mistake: Klaus tells Steve how he can take him to ASU and introduce him to some choice people. Steve replies "I don't know, isn't that a party school?" and acts like he doesn't want to attend. However, in an earlier episode "A Nice Night for a Drive", Klaus tells Stan that Steve is hoping to attend ASU because of "sluts on campus."
Beyond the Alcove or: How I Learned to Stop Worrying and Love Klaus - S17-E7
Continuity mistake: When Francine is arrested for being a menace, one of the police officers does not have a badge over his shirt pocket, but after Francine vanishes in the cloud of smoke, a badge suddenly appears on the shirt. (00:16:35)
Chosen answer: It is just the decision of the creators to do so.
MasterOfAll