Continuity mistake: Brian gets the fur on his hands burned off by the kitchen fire, but when Lois goes crazy and goes into the living room, the rest follow her and the screen zooms out. When that happens Brian's fur is back on his hands.
Continuity mistake: When Stewie is in his room talking to Rupert and himself out loud about Janet we see that Rupert is on the left side of Stewie's pillow. When the camera pans over to Brian who overhears Stewie's musings, after Brian makes fun of Stewie the camera pans back and Rupert is gone. There is also a quick tete-a-tete on the color of Janet's eyes and the final shot is on Stewie and his crib and yet no Rupert to be found.
Continuity mistake: During the end of Peter's "revised" version of "The King and I" right after the scene where Brian and Peter kung-fu fight, the scene transitions to the next scene where a bunch of girls with bikinis on are surrounding Peter. During the transition, a girl appears out of nowhere. She is blonde and is leaning on Peter.
Continuity mistake: When Simon is working with Stewie in his "The Life Cycle" exercise, there are three children behind him: a redhead with freckles, a blonde with glasses, and another blonde. However, after Simon turns around and says, "Any comments?", there is a wide pan of the kids in the class, and the three kids have completely disappeared.