Continuity mistake: When the family goes to Morning Wood Academy, the sign in the front says No Irish and Irish OK underneath that. The next time we see the sign, those words are replaced with the date the academy was founded.
Whistle While Your Wife Works - S5-E5
Other mistake: When the guard is turning the vault handle, the handle doesn't move, but the guard's arms do.
Barely Legal - S5-E8
Audio problem: Just after Peter graduates from the Police Academy, he goes home to Lois and draws his gun in the kitchen, which gets Lois very excited. Every time Peter fires his revolver, you'll hear a soft, but distinctive clink briefly after each shot. This clink comes from shell casings hitting the floor, which revolvers make impossible. Revolvers are designed to eject expended shell casings only while its user is reloading, not after each shot is fired. Only semiautomatic firearms immediately eject expended cartridges immediately after having been fired. Unless I'm mistaken, the entire gunshot sound originally came from a 9mm semiautomatic pistol.
Character mistake: In "The Drunken Clam", while Quagmire and Joe are playing table tennis, a poster on the wall incorrectly spells "Pawtucket Patriot ale" as "Pawtucket Patrtot ale."
Continuity mistake: When Lois opens the door to Peters mum, Peter appears at the door and Lois simply vanishes. (00:05:00)
Continuity mistake: When Lois is talking to Brian about having homicidal thoughts, the rug colours swap order. (00:14:20)
Continuity mistake: At the start, Chris' hat is all yellow instead of yellow and black. (00:00:30)
Continuity mistake: Lois and Brian go outside and celebrate Peters dads death. Behind them you see the trash cans. They come back in and sit down and the trash cans have vanished. (00:12:33 - 00:12:54)
Continuity mistake: When Meg's boss tells her to fire Peter, he has plungers on his nipples. He walks over to Meg and takes them off. A second later his shirt is tucked in without Peter being able to tuck it in. He's holding plungers in both hands.
The Tan Aquatic with Steve Zissou - S5-E11
Continuity mistake: Stewie uses a chair to climb into his tanning bed. He tells Brian to come get him in 15 minutes. When he comes back 6 hours later, the chair is back over by the table.
Revealing mistake: As Stewie is walking toward Herbert's house, he just suddenly appears mid-screen.
Revealing mistake: Right after the "Ghost Whisperer" cutaway, while Peter, Chris, and Lois are sitting on the couch, Chris' eyes disappear for a single frame.
Continuity mistake: The gun in Peter's first cutaway vanishes when the stage goes dark.
Continuity mistake: One of the ping pong balls that fall off Peter vanishes when it hits the ground.
Barely Legal - S5-E8
Revealing mistake: Every time Cleveland's tub slides out of the house, there's never any plumbing attached to it. It just sits on top of the floor so it slides out easily. In this episode its very easy to see.
Continuity mistake: As the helicopter is shown sliding down the hill, it is shown from several angles to have just a single rotor blade. However, one shot shows two rotors, one of which disappears halfway through the shot.
Boys Do Cry - S5-E15
Revealing mistake: When Peter brands the cow, no burn mark is left on the cow. (00:18:20)
Boys Do Cry - S5-E15
Continuity mistake: At the Little Miss Texas pageant, just as Lois finishes saying "It's what you do when you're in Texas", she and Stewie walk past a judge and a little girl. In a close-up of Stewie, the judge and little girl have suddenly become a plant. (00:15:25)
Continuity mistake: Peter says "Oh crap" as soon as he realises Ernie the Giant Chicken is looking at him in the window. When the camera pans over to Ernie, he is standing an inch away from the glass. It then cuts to a wide angle where Ernie jumps through the window, and he has instantly moved back a few inches. (00:07:00)

Continuity mistake: In the restaurant scene during the chicken fight, after Peter says "oh yeah? What are ya gonna do about it?" the chicken already has a black eye just before they begin fighting.
Answer: Mort is Jewish, and Walt Disney was said to be antisemitic. So it would make sense in a Disney universe all the inhabitants would share his supposed beliefs.
MasterOfAll