Continuity mistake: Just before Peter enters the tunnel in his new car, the white line in the middle of the road is blotchy and not continuous. When the shot changes to Peter's POV, the line has completely disappeared. The next shot which shows the car side on, there is now a straight white line on the road.
Continuity mistake: In the episode "The Cleveland-Loretta Quagmire" while the Griffin family are at the Heimlich maneuver class, there is a scene where Stewie is talking to a doll and putting his clothes on. You can see that he has no shoes on but when he sticks his feet through the pant leg holes, shoes suddenly come out of the pants and he has his shoes on.
Continuity mistake: Stewie forces Lois to breastfeed him, and she undoes her shirt. As Brian and Lois argue about who is going to test her milk, Cleveland barges in and is shot. When Lois jumps up, her shirt is buttoned again, with no time for her to do it.
Continuity mistake: Stewie tries to get his mind control device back from the kitchen. The cupboard door is open when he falls. In the very next shot, Lois turns the light on in the kitchen, and we see the door has closed without anyone touching it. It is open again after that.
Continuity mistake: While Stewie and Olivia are performing, Olivia is wearing a light purple dress. After the performance, when Stewie and Olivia are sitting on the piano talking to their teacher, Olivia's dress changes from purple to blue in between shots. She is also wearing dark red lipstick instead of none at all.
Continuity mistake: In the scene where Stewie is eating chocolate cake, he shoves his head in it and covers himself in cake. However when Peter walks in after having liposuction, all the cake has disappeared and Stewie is now completely clean. The bowl of grapes on the table is also gone.
Deliberate mistake: In season 4 episode 7, 'Brian the bachelor'. Cleveland is sitting in a chair in a waiting room and is getting really nervous. Peter tries to help him by tying him to a chair and when that doesn't work, he takes off all of their clothes. Peter ties Cleveland's hands to the arms of the chair and then takes Cleveland's clothes off. He couldn't have taken the shirt off while Cleveland's hands were tied to something without untying him or ripping the shirt, which he didn't do.