The One Where Monica Gets a Roommate - S1-E1
Other mistake: When Rachel arrives at Central Perk, Monica introduces her to everybody, including to Chandler, as if they don't know each other. In later episodes, we learn that Chandler met Monica and Rachel a couple of times when they were in high school (including when Monica cut his toe off). Chandler even kissed Rachel once, and since Chandler always says that he hasn't kissed many women, he should have perfectly remembered Rachel, and Monica should have remembered that the two of them knew each other. (00:04:40)
Suggested correction: Chandler and Rachel never really knew each other well. Chandler will have been in her mind as that guy that her best friend's brother hangs out with, and Chandler will only know her as the friend of his friend's sister. I doubt they'd remember each other as the kiss they shared was when she was drunk, and they didn't see each other for 6 years while Rachel lived a richer and dependent on her father type life.
The One with the Dozen Lasagnas - S1-E12
Revealing mistake: After Phoebe tells Rachel that Paolo made a pass at her, we see a shot of the outside of Monica's apartment block and Rachel is throwing Paolo's clothes out of the window. Look closely and you can see the clothes have been very badly superimposed on to the background. (00:17:05)
Suggested correction: The show sent a crew to New York to film the clothes being thrown from the same building that is used in establishing shots. In a wide-screen version of the episode, the clothes can be seen being thrown out a window (although in the episode the clothes are supposedly thrown off a balcony).
Continuity mistake: After Chandler makes the joke about Joey cracking show business, Joey is holding the phone in his left hand. In the next shot, it is in his right hand. (00:11:20)
Suggested correction: You can actually see him switch it to the other hand before the next scene pops up if you look close enough.
Not for the shot the mistake is talking about. Throughout the scene he's switching the phone back and forth. But after Chandler's line, Joey is just standing there with the phone in his left hand, he doesn't make any move to switch hands. In the next shot, the wide shot from the side showing everyone, you see the phone is in his right hand.
The One Where Underdog Gets Away - S1-E9
Continuity mistake: At the end of the episode when Chandler makes a toast, he is holding the glass in his right hand. In the next shot, the glass has jumped to his left hand. (00:21:05)
Suggested correction: He switches the glass from right to left, so, unfortunately you're wrong.
The mistake is valid. While we do see him switching his glass between hands during the scene, when he says he's glad everyone's Thanksgiving sucked, he raises his glass in his right hand. In the next shot it's switched to his left and lowered a bit. Since this happens between shots (and he's not off camera), it means no time elapsed, so there's no time for him to switch hands again (in the submitted pictures, the bottom picture is the first shot and the top picture is the 2nd shot).
The One with the Thumb - S1-E3
Continuity mistake: Monica's snort, Joey's knuckle cracking, and Phoebe's hair chewing are never seen in any other episode, before or after this one.
The One with the Blackout - S1-E7
Audio problem: When Phoebe is counting down to the last candle burning out you can see Joey blowing out the candle, but the sound of his blow is only heard a second later after he sits back. (00:22:50)
The One with the Blackout - S1-E7
Other mistake: Phoebe says she was last to know Monica had a crush on Joey when he moved in and Joey acts as though he never knew this. However, when we see him moving in a later episode, he knows Monica has a crush on him, and he is under the impression that Monica wants sex and so strips off in the living room while Monica is preparing some drinks for the pair. (00:13:10)
Suggested correction: I may understand the situation but please do correct me if I'm wrong. Basically in the flashback episode they saw each other in the hallway and smiled at each other. They would've both got the impression that the other one liked them. However, after the lemonade incident he would have then got the impression that he was wrong and Monica didn't actually like him. Finally, Joey heard that she did like him and got a bit confused and a bit happy at the same time.