Continuity mistake: When Rachel and Phoebe are talking in the Baby Beauty Contest and Rachel is holding Emma, Emma's head instantly changes position between shots several times, often going through 180 degrees. Clearly they couldn't get her to do the scene in one take and therefore the editing suffers.
Revealing mistake: At the point in the ceremony where Joey forgets the rings, the camera pans out and you see Chandler step sidewards to get the ring from Ross, and Monica step sidewards to get the ring from Rachel and Phoebe. As Monica moves, her dress slides across and one of her feet is revealed; it has what looks like a brown boot or heavy brown shoe on it.
Continuity mistake: During the scene where Phoebe is telling Rachel that Paolo made a pass at her, Rachel unrolls a long length of kitchen roll. During the scene there is a shot over her shoulder where we see that the kitchen roll is as normal and in the next shot we see it unrolled again.
Revealing mistake: When Phoebe says something like, "Look, the last candle is about to go out", she starts counting down, saying "-47,-48..." and you can see Monica looking at the candle. But there is a light source behind her (it can't be a candle because she is looking at the last candle).
Continuity mistake: Joey goes into the bathroom and the toilet is behind the door. In a previous episode, when Joey doesn't know about Chandler and Monica, he comes out of his room and thinks it is morning. He goes into the bathroom and Chandler pushes the door open to reveal Joey asleep on the toilet with his toothbrush in his mouth, only the toilet is now on the other side of the room, not behind the door.
Continuity mistake: Paul finally opens up, and cries like a baby to Rachel. While talking to her at the breakfast bar in the kitchen (when Rachel is trying to distract him with sex), he's wearing a white linen shirt that constantly changes between tear-stained and wrinkled to ironed and clean.
Continuity mistake: In the one where Ross finds out he's having a boy (series 1), just before Rachel lets it slip, she is sitting on the edge of a chair eating a cookie. Watch her dress, it changes back and forth from being undone from the waist up, to only having two buttons open at the top.