The One With The Proposal, part 1 - S6-E24
Factual error: There's no charity event in the world that would be clamoring for non-paying guests and offering them free food and an open bar.
The One Where Rachel Quits - S3-E10
Continuity mistake: In the very first shot of Chandler and Rachel sitting on the couch, Chandler has his legs crossed left over right, but it cuts to the closeup and his legs are crossed right over left, then it's back to left over right.
Factual error: The makeup on Rachel's face "can't be covered up with makeup," and "won't come off," we learn in The One In Vegas, Part 2. So it's not a surprise that she still wakes up with it on (and so does Ross.) but an hour or so later, when she's at breakfast, it's not there. An hour or two won't make the magical difference that 8+ hours didn't.
The One Where Joey Moves Out - S2-E16
Other mistake: When Rachel is showing Phoebe her heart tattoo, the tattoo isn't dressed and covered, nor is the skin around it red or swollen, despite the fact she just finished getting it. (00:14:09)
The One Where Joey Dates Rachel - S8-E12
Factual error: When Chandler is explaining that he played Ms. Pac Man all day, the Ms. Pac Man machine is unplugged, and then plugged back in. Immediately upon being plugged back in, we hear the Ms. Pac Man theme song being played. No version of a Ms. Pac Man stand-up arcade ever played the theme song upon powering on. Instead, it would silently go into "demo mode." The only way the stand-up arcade version of Ms. Pac Man would ever play its theme song was after two events occurred in order: 1. A coin was dropped in the slot (or a button was pressed on the inside of the unlocked coin drawer), and then 2. Someone pressed the '1 player' or '2 players' button to start the game. It is clear that neither of these things has happened in the scene by the time the music starts playing. (00:11:50)
The One With The Home Study - S10-E7
Other mistake: In the opening scene Chandler smiles at Ross' joke while playing with the ball in his hands. Later on in the scene Ross cracks another joke and the exact same scene of Chandler smiling and fiddling with the ball is repeated. (00:00:40 - 00:01:10)
The One With The Stain - S8-E7
Continuity mistake: When Monica asks Brenda to clean the top of the cabinets so that she can get a look at her crotch, Brenda drags a chair over (with both hands) to stand on. In the next shot, she is magically holding a feather duster. (00:11:40)
The One Where Ross Got High - S6-E9
Plot hole: Monica is trusting Rachel to make a traditional English trifle and being a top chef (or an even averagely intelligent human being) she would know what ingredients are used in one. So there's no way she could have missed Rachel cooking beef with peas and onions during the cooking process. Rachel's cooking it right there in Monica's kitchen while everyone else is there!
The One With Mac & C.H.E.E.S.E. - S6-E20
Continuity mistake: In the episode before, Rachel's hair is just underneath her shoulders. Then, a few weeks later when Ross gets back from Spring Break, it's a lot longer than before.
The One Where Rachel Tells... - S8-E3
Visible crew/equipment: When Monica is yelling at the other couple on their honeymoon, the things on the shelves behind them shows the reflections of set lights. (00:15:25)
The One Where Old Yeller Dies - S2-E20
Continuity mistake: A woman and a man in a blue shirt exit the coffee shop twice.
The One With The Bullies - S2-E21
Continuity mistake: A woman and a man in a blue shirt exit the coffee shop twice.
The One With The Bullies - S2-E21
Visible crew/equipment: While driving the cab, Phoebe is wearing a blue shirt. When the car stops suddenly, a crew member wearing a white shirt is visible driving the car.
The One With The Halloween Party - S8-E6
Audio problem: When Rachel greets Mona at the door of the apartment, visually she says "Hi" but the audio is " Hi Mona. "
The One With Chandler's Dad - S7-E22
Character mistake: When Chandler says "and there's daddy", an extra behind him chuckles in time with the studio audience. He's the only one laughing and it happens right on cue with the audience.
The One Where the Monkey Gets Away - S1-E19
Continuity mistake: When Rachel tells Ross that she "kind of lost" Marcel it cuts to an exterior shot from the balcony, and if we look past Rachel and Ross we see there's a table against the wall, under the Maïna La Voyante poster, but that table does not exist in previous or following shots. (00:10:15)
The One With Joey's Bag - S5-E13
Continuity mistake: When Phoebe realises she just met her dad she turns to her friends to share the news, and in the closeup she's holding a pair of 3D glasses in her right hand and Monica's holding a pair in her left hand, but in the next wideshot both of those glasses have vanished. (00:09:15)
The One With Joey's Bag - S5-E13
Continuity mistake: When Chandler speeds up the timer for his turn he places the timer down at the right side of the pillow, but it's at the opposite side of the pillow when he lies down. Then before Monica starts giving Chandler his massage, she sets the timer and places it near the foot of the bed which we see in Chandler's closeup, but in the next shot when she says, "Say goodbye to sore muscles," oblivious to Chandler's pain, the timer is lying by the pillows behind her, near the head of the bed. Note, in the "uncut scene" version, Chandler grabs the timer to speed up his torture, and Monica takes it out of his hand thinking he was going to add more time, then she places the timer behind her. (00:00:25)
The One With The Apothecary Table - S6-E11
Audio problem: When Phoebe puts her feet up on Ross' apothecary table, Ross asks her not to put her feet up on his "new...old sheet." During the pause right after he says "new," there's a cut to a different angle and you can see his mouth form the words "old sheet." Then it cuts to a different angle and we actually see and hear Ross say "old sheet." (00:12:05)
The One With All The Jealousy - S3-E12
Continuity mistake: At the start, Ross says, "Botanists are such geeks." When he does, his left hand goes from in his lap to on the table instantly. (00:00:35)
Answer: The painting doesn't have a name, and if it does it is never mentioned in this episode.