The One Where No One's Ready - S3-E2
Continuity mistake: When Phoebe comes out of the bathroom after trying to get out the hummus stain on her dress, the bathroom door is left wide open. But when Monica is playing her message, the door is suddenly almost shut, though no-one had gone near the door.
Audio problem: In the scene where Malcolm is talking to Phoebe in the subway, you can hear him but his lips aren't moving.
Audio problem: When Rachel and Ross are in Central Perk, Ross says "Well there's this one thing." Then it cuts to a shot of the side of his head where he says "Okay" and it doesn't match his lips.
The One With The Cheap Wedding Dress - S7-E17
Continuity mistake: In the shot where Joey meets Kristen the lights in the store window behind them are switched on. On the DVD at 05:19, you can see that the "CD" light is illuminated in blue. However, at 05:36 where they both shake hands, all the lights are switched off. (00:05:15)
The One With Rachel's Other Sister - S9-E8
Continuity mistake: When Chandler puts the box of china down on the chair, the lid of the box is shut, but a bit later the box is wide open.
The One With The Ballroom Dancing - S4-E4
Revealing mistake: During the end bit, before we get to see Phoebe when she is being interviewed at Lara and Jeni's Massage, if you look closely at the name of the massage parlor, you'll see that, although the camera shakes and bumps, the words stay on the very same spot on the screen. Obviously the canopy was blank and the name was digitally added later.
The One With The Mugging - S9-E15
Visible crew/equipment: When Chandler slips while wearing the shoes, if you look above Monica and Rachel, you will see a stage light.

The One with the Boobies - S1-E13
Continuity mistake: When Joey's mom comes to tell him that she knows about the affair, he places the paper bag that she brings on the stove and it falls over. In later shots, it is obvious that the bag is upright. (00:17:05)
The One With The Home Study - S10-E7
Continuity mistake: When Chandler kisses Monica before letting the social worker in, she puts her hand on his arm. The camera cuts to her and she says "Tongue, really?" but her hand is by her side now. The camera cuts back to Chandler and both her hands are now out in front of her.
The One Where Rachel Has a Baby, part 2 - S8-E24
Visible crew/equipment: In the very last shot of the episode, as Ross is walking down the hall, stage lights are visible at the end of the hallway.

The One Without The Ski Trip - S3-E17
Continuity mistake: When Monica and Phoebe enter the coffee house and talk to Chandler and Ross, the extra sitting at the coffee bar (black shirt, blue jeans) is not wearing a belt, then when they pan again she is, and back she is not. (00:00:12 - 00:00:54)
The One With The Ride-Along - S5-E20
Plot hole: When Rachel picks up the box of photos they all fall out, because the box isn't sealed with tape or even folded properly. But there's no way for that to have happened. Monica would have brought the box from somewhere and put it on the couch, and the photos would have fallen out then, and she would have fixed it (this is Monica, after all.) The only way for this to have happened is if she put the empty box on the couch, unfolded/unsealed, and then put photos in it - which Monica would not do.
The One With Chandler's Dad - S7-E22
Continuity mistake: When Ross is in the car with Rachel he sat in the back you can see him he has his arms crossed and then when the camera cuts to the front view he is holding the seat in front. (00:04:50)
The One With Joey's Big Break - S5-E22
Other mistake: When Ross is questioning Phoebe on why she is mad at him, he has his blue sweater on backwards.
Suggested correction: So he has his sweater on backwards. That's a character mistake, not a movie mistake.
The One With The Halloween Party - S8-E6
Other mistake: When Monica is saying "I can't answer that, Chandler's my husband" after Joey asks her who would win in a fight between Ross and Chandler, the camera is fluttering in and out of focus. (00:11:11)

The One With Joey's New Brain - S7-E15
Visible crew/equipment: When Monica and Chandler are going over to the window to see Ross playing the bagpipes, the head of a crew member is visible on the left side of the screen, standing by the framed poster. (00:07:30)
Character mistake: In some episodes the facade of Central Perk is shown. On the window the word "cappuccino" is misspelled, it says "cappucino."

The One with the East German Laundry Detergent - S1-E5
Character mistake: In the laundromat, there is a sign where the word clothes is misspelt cloths. (00:16:00)

The One with the Candy Hearts - S1-E14
Visible crew/equipment: Just as Joey and his date are leaving, a stage light and the top of the set drop into view above Chandler. (00:08:10)
The One with the Dozen Lasagnas - S1-E12
Audio problem: The very last line of the episode (Monica saying "and I'm gone") is an overlay. (00:22:40)