The One Where They All Turn Thirty - S7-E14
Plot hole: There's no way Phoebe has lived to 31 without knowing her birth date. She's had "real" jobs (like the massage parlor) and those require social security numbers and accurate birth dates. She didn't go to high school, but she mentions classmates, so she must have gone to elementary/middle school, and those required birth dates even in the '70s and '80s. She's also "lived in Prague," which she couldn't have visited without a passport - which requires a birth certificate.
The One with the Blackout - S1-E7
Other mistake: When Ross and Rachel are out on the balcony alone for the first time, there are no lights on in NYC apart from candle lights. Look in the BBQ behind them though to see a very bright and very white light reflected in the paintwork. (00:13:50)
The One With All The Kissing - S5-E2
Continuity mistake: A few days after the friends have returned to New York following Ross' wedding, Monica's hair is suddenly about three inches longer and a completely different style than it was in London. It was previously just above her shoulders in a sleek "pageboy" style. Once back home, it is much bushier and longer, well over her shoulders. This was obviously due to Courtney Cox growing her hair during the break in-between Season 4, which ended with the Ross/Emily wedding debacle, and Season 5. (The first episode of Season 5 was filmed the previous season and was Part 2 of Ross and Emily's wedding).
Character mistake: Even though both Ross and Rachel have cell phones with them and use them for other calls, it never occurs to Ross early on and as he's frantically racing to the airport with Phoebe, to call and tell Rachel that he loves her or at least to say that he is on his way to tell her something important before her plane takes off for Paris.
The One Where Ross Got High - S6-E9
Continuity mistake: When Rachel tells Phoebe that she's making a traditional English trifle, Monica's holding a cloth napkin, but in the next shot the napkin's suddenly gone and she's now holding forks.
Other mistake: In "TOW the cheap wedding dress" the whole episode is about how Chandler wants "The Swing Kings" to play at the wedding, because, at their concert, that is where he realised that Monica was the woman he wanted to dance all his dances with, and Monica goes to great lengths to secure the band. In this episode, the band at the wedding is actually called "Jungle Swing".
Suggested correction: So the band changed their name in the interim. It happens.
The One With Ross's Wedding, part 2 - S4-E24
Audio problem: When the wedding officiant says "take thee, Ross" his lips don't move. (00:20:35)
The One Where Ross Got High - S6-E9
Continuity mistake: When Phoebe finishes telling Rachel about her thanksgiving dream about Jack, Chandler is looking straight ahead with his arms crossed. Cut to the next shot when Jack and Judy are coming out of Monica and Chandler's room, and his arms are immediately uncrossed and he's looking at the women. (00:20:05)
The One With Phoebe's Dad - S2-E9
Continuity mistake: When Ross gives the super the money and asks "will this help with the knob" the super says that the place isn't open until Tuesday. When he says that the money is in his right hand. Then when he asks "am I not saying it right", the money in his right hand is gone.
The One With The Home Study - S10-E7
Continuity mistake: When Mike says "I am going to make this decision for us. What do you think we should do?" he is holding the check. The very next shot shows the charity worker taking the check off the desk and giving it back even though Mike never put it down. (00:20:35)
The One With Joey's Fridge - S6-E19
Revealing mistake: In the closing scene where Ross and Elizabeth are dancing on the beach, you can see that the sky in the background is actually a picture, and you can see where it ends. (00:21:20)
The One with the Sonogram at the End - S1-E2
Continuity mistake: Toward the end of the episode, the exterior shot of Central Perk shows about 8-10 people inside the cafe, at the coffee bar, at the tables. But the next interior shot shows the cafe is empty except for our six Friends, lounging in the closed cafe while Rachel cleans up. (00:12:15)
The One In Barbados, part 2 - S9-E24
Character mistake: When Mike and Monica are playing ping pong, Mike's service was a fault as he released and hit the ball over the top of the table rather than behind. He then (on the same point) claims to win the point because Monica placed her hand on the table. (00:33:00)
The One In Barbados, part 2 - S9-E24
Continuity mistake: In the scene where David wants to propose, when Phoebe and David get seated in the restaurant Phoebe puts her bag on the floor to the left of her chair. When there is a wider shot again, it has vanished. (00:25:25)
The One With the Sharks - S9-E4
Continuity mistake: When Ross explains to Phoebe about "Vikram" the big dark jewel on her necklace switches repeatedly from hanging at the left side of her neck to the middle, then to the right side and again to the left side again between shots. (00:16:10)
The One Where Monica Gets a Roommate - S1-E1
Audio problem: The camera cuts briefly to Chandler as the friends are discussing Monica's love life. During this very brief time, you can see Phoebe's mouth moving but there is no audio of her voice, and Monica's voice is heard talking in the background instead. (00:01:10)
The One With Mac & C.H.E.E.S.E. - S6-E20
Other mistake: When we see the chess board at Central Perk that Ross and Chandler are using, the black pawn in front of the black knight nearest the camera is missing, and from the arrangement of the white pieces, it cannot have been captured yet.
The One With Rachel's Phone Number - S9-E9
Visible crew/equipment: In the scene where Chandler pretends that he just got home from Tulsa and meets Joey who is waiting in the hallway outside the apartment, the camera cuts between Chandler and Joey. Every time the camera shows Chandler, there is a wooden wall at the left edge of the screen with an orange tape. This is evidently part of the set, because when the camera cuts to Joey, the wooden wall with the orange tape is nowhere to be seen. (00:16:45)
The One With Ross's Inappropriate Song - S9-E7
Continuity mistake: When Ross is holding Emma, the position he is holding her changes.
The One With The Joke - S6-E12
Continuity mistake: In the coffee house scene when Joey reveals that he is working there, the strap on Joey's apron goes back and forth from being twisted around his neck and then straightened out. (00:10:50 - 00:12:50)