Continuity mistake: When Stewie is golfing and spots Brian in disguise, Stewie has a putter. When Stewie realizes that it is Brian and goes after him he is holding a chipper.

The Thin White Line (1) - S3-E1
Continuity mistake: When Brian is saying that his family has it all wrong about his drugs, Lois is holding Stewie. In the very next shot of his family from the front, Stewie is standing on the ground, without any time for him to climb down. (00:15:10)

I Never Met the Dead Man - S1-E2
Continuity mistake: When Peter is explaining to Meg about "bribes" in one shot the window in the kitchen door disappears. (00:09:00)

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. (00:14:20)

One If By Clam, Two If By Sea - S3-E4
Continuity mistake: After Peter has been to the newly opened bar, he runs to his house door (which is red) and when he opens it, the colour has changed to brown.
Continuity mistake: During the Star Wars rip-off at the end, just before Peter climbs the Town Hall stairs, the people on the sidelines change. (00:19:45)

Peter Griffin: Husband, Father...Brother? - S3-E14
Continuity mistake: When Stewie jumps on the picnic table and begins his dance, at one point he undoes his coveralls and removes his shirt. In the next shot he is fully clothed.
Chick Cancer - S5-E7
Continuity mistake: Just before Peter announces his decision to make a chick flick, the tissues on his lap change position.

I Never Met the Dead Man - S1-E2
Continuity mistake: When Peter crashes into the cable tower in the far shot, the number plate isn't there. Yet in the close up shots it is there. (00:06:05)

Continuity mistake: At the football game, the row of people behind the Griffins disappear in one shot, and reappear in the next, with not enough time for them to have all moved away, and all moved back. (00:07:10)

Continuity mistake: When the government people are looking at the missiles spelling 'Happy New Year', the 'Y' in 'Year' changes distinctly between shots. (00:06:50)
Peter, Peter, Caviar Eater - S2-E1
Continuity mistake: In the scene where Lois and Peter are at the historical auction, when Lois stops to talk to her friend the person standing next to her friend is wearing a green suit. In the very next shot he is wearing a black suit and vest.
Peterotica - S4-E24
Continuity mistake: After the Quahog Creek clip is over, they show a picture of the house. In the background there is a tall silver building that wasn't there the last time they showed a shot of the house. (00:17:45)
There's Something About Paulie - S2-E16
Continuity mistake: When Lois asks to speak to Peter in the other room after sitting down for supper with Paulie, the door they come out of is where the front door is all the time, not the kitchen.

Continuity mistake: When Chris hides the vodka from his teacher, he holds his right hand behind his back with his left arm at his side, but in the next shot, both his hands are behind his back. (00:09:40)
Continuity mistake: When Stewie has chicken pox the red marks on his face change position many times between shots. A noticeable one is just as he is saying "No, damn you, I've got to finish what I started".
Mr. Griffin Goes to Washington - S3-E3
Continuity mistake: When Mr. Weed is crying about losing his job, we see a view of a long line of trees in the window behind him. The camera switches to a close-up of Peter. When it zooms out again, the trees on the right of Mr. Weed are missing.
Suggested correction: I watched this part of the episode twice and no trees are missing in the window behind Mr. Weed after the close-up on Peter. Actually, Mr. Weed's head is covering part of the window where you can't see the trees on the right when he's lying on the table, but when he gets up you can see all the trees.

Continuity mistake: In the scene where the family are singing an African song, they all point their arms up to the ceiling yet in the next shot, Lois's arms are immediately cupped and pointed downwards - you don't see her moving her arms to that position.

Continuity mistake: When Joe arrives at the orphanage when he's telling his story, the sign changes position and the writing changes on it. (00:11:30)
Continuity mistake: Stewie and Brian shoot themselves in the foot. Next shot the holes have disappeared.