Continuity mistake: In the opening titles (for series 1-3), Stan runs outside and picks up a newspaper. He reads the headline and throws it back on the ground. But by the time Stan drives off, the newspaper has disappeared. (00:00:10)
Continuity mistake: At the convention when we first see Roger and Hayley, they pass a booth that says Costume contest registration. There is also a few people around, a poster and a movable wall. The very next shot is more of a close up of the two and the background has changed dramatically. There is now about 4-5 different stalls behind them and no sign of the previous one or the poster and wall.
Revealing mistake: When Stan comes home and starts talking to Francine, as Francine is saying "You're just having a mid-life crisis", there are three objects in the background that are cut in half and seem to be floating in the air. (00:02:10)
Continuity mistake: When Francine is placing a breakfast plate on the table, the arrow on the fridge's temperature dial is pointed to the middle of the yellow. Two shots later, the arrow is pointed to edge of the blue and yellow. (00:00:50)
Continuity mistake: The window partway up the Smith staircase vanishes after the first season.
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)
Roger Codger - S1-E6
Continuity mistake: In the Area 51 flashback, the "RESTRICTED" sign on the lab door goes from red to blue between shots. (00:11:00)
Tearjerker - S3-E10
Character mistake: When Tearjerker is describing his evil plot to Stan, he says he auditioned for a role in Monster's Ball in 2002. Monster's Ball was released in 2001.
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)
Weiner of Our Discontent - S4-E18
Continuity mistake: When Rodger comes to the table, he takes a waffle from the plate, starts eating it and then puts it on the plate. When Steve enters the room and sits down, the plate goes from no waffle to waffle again.
Continuity mistake: In the pilot episode, during the breakfast scene, there are five donuts in the box in front of Roger before his chair breaks. When he takes one from the box, there are eight. When Francine takes his donut away, there are five in the box again. (00:03:20 - 00:03:50)
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)
The Fast and the Spurious - S17-E13
Continuity mistake: When future Roger announces his upgraded seatbelts, the way some of the double seatbelt straps are crisscrossed changes in following shots. (00:16:55)
Phantom of the Telethon - S4-E7
Continuity mistake: The way Greg holds the volleyball at the start changes instantly. (00:02:00)
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.
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.
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 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."
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)
Chosen answer: It is just the decision of the creators to do so.
MasterOfAll