When Joey is counting the floors of Ross' building (the first time) to find out where Hot Girl lives, he is wearing a black top with a zip. When he rushes over to Ross' building he walks down the hall and knocks on what he believes to be Hot Girl's door. In these shots he is wearing a dark purple button up shirt. As soon as Ross answers the door, Joey is wearing the black top with a zip again.
When Rachel finds out that Ross and Julie are getting a cat, watch Rachel's neck. When she says "together?" she's not wearing a necklace; cut to Ross and Julie for a fraction of a second, then when it cuts back to Rachel saying "both of you?" she's suddenly wearing a necklace. It then disappears again when she says "isn't that just lovely?"
The number of the flats near the start of the first series was 4 and 5. After a few shows they changed to 19 and 20. [The producers realised that the numbers 4 and 5 were for apartments on the lower floors and had shown Monica's apartment to be much higher up so they changed the numbers - it's still a mistake but there's the reason.]
When Joey goes to Monica and Chandler's apartment, he goes to the fridge to get something to drink to keep his bladder full for the audition. When he closes the fridge he has two bottles in his left hand. The camera cuts to Monica, then cuts back to Joey. He then has two cans in his left hand and is holding them differently.
Chandler and Joey are spending time with Richard; as they play foosball and Richard scores, he leans back and says "Could that shot BE any prettier?" in a mockery of Chandler. The end of the wall and the hallway behind it are visible (ie the wall set ends).
Monica is sitting in the living room with all her wedding presents, waiting for Chandler to return. She begins to open a large parcel and reveals a brown cardboard box. Then the camera moves to Joey, and when it moves back, the parcel is wrapped up again. This happens about three times.
When Phoebe goes to visit her father, there is a little dog outside. Phoebe gets back in the car and the dog jumps up the side of it. When he finally gets part way through the window, the top of the dog-handler's head is visible through the car window.
Towards the end, during a scene at Ross and Rachel's, Sandy picks up two puppets and puts them on each hand. He starts pulling the blue one onto his left hand, but when the angle changes there's a yellow one on his left hand, and a blue one on his right. It cuts to Ross, then when we go back to Sandy they've changed hands again.
When Monica is waving around her shellfish hairdo in the final scene at the hotel, the camera zooms out. When it does this, look at the top of the set. Not only can you see where the set ends, but you can also see the stagelights, hanging equipment etc.
Monica is talking to Pete in the diner when he gets back from Japan. If you look at the floor at Monica's end of the counter at the very beginning of the scene, you can see the mat that is there to protect her when she falls over later.
Just before Ross gets hit in the hockey game, you can see him wearing a foam finger which then disappears.
In the scene where Rachel is telling Ross it is sexy for a man to wait before having sex, the cups behind Ross change color: red to green to red and back again.
In the scene where Joey's teaching soap opera for actors, his name changes from the 'a' being on the line to being above it. The line itself also reaches the far right of the board in some shots, and is short of it in others.
Monica and Richard are squishing tomatoes; when she starts dabbing his shirt with club soda the door behind them leading to Monica's room goes from open to shut between two shots.
In the scene after Monica pretends to drop socks in the hallway in order to prevent Rachel from telling Ross she still loves him, Rachel goes back into the apartment and locks Monica out. Monica then starts to pound on the door with her bare left hand. The scene changes to the apartment and Ross opens the door to let Monica in. All of a sudden Monica is pounding the door with her right hand wrapped in a sock.
When Joey refuses to give Chandler the chair, Phoebe gets hummus on her dress, but in some scenes the stain is much smaller than others.
When Chandler and Monica are at the hospital holding the twins right before they try and swap babies, there are two hospital tags around Chandler's left wrist which then disappear before re-appearing.
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.
The food/sauce stain Joey gets on his shirt (after running into the cart) changes between shots.
When Rachel first meets Paolo she's looking for the cat in candlelight, holding the candle in front of her. You can see her shadow on the wall, as well as the shadow of the candle, revealing that the light is not coming from the candle.