The One with the Blackout - S1-E7
Continuity mistake: During the Monopoly game (deleted scene), right after Monica says, "we can bring my wheelbarrow, to carry your teeth," we see Ross reading the game manual at the end of the shot. At the beginning of the very next shot, he is suddenly rolling the dice with no manual in his hands. (00:17:00)
The One with the Birth - S1-E23
Continuity mistake: In the scene when Carol is giving birth, there is a shot when Carol shouts "Where were you?" Her trousers are visible under the sheet. (00:16:25)
The One with the Blackout - S1-E7
Continuity mistake: After the light comes back on and Ross is holding his heart after he sees Rachel kissing Paulo, in the next shot when Joey starts speaking to him his hand is by his side. (00:20:40)
The One Where the Monkey Gets Away - S1-E19
Continuity mistake: When Ross comes back to Monica and Rachel's apartment and Rachel tells him she has lost Marcel, he takes off his overcoat and lays it over the back of the sofa and then sits by it. As the scene continues, the sofa is still visible in some shots and the black coat is crumpled up on the sofa cushion. (00:11:25)
The One with Two Parts: Part 2 - S1-E17
Continuity mistake: When Joey is talking to Ross and Chandler about missing Phoebe's birthday, Chandler has the newspaper in his hands. In the next shot, you can see that Chandler does not move at all, then in the next shot, the paper is on the table. (00:04:35)
The One with All the Poker - S1-E18
Continuity mistake: In the scene where Rachel and Ross are playing poker head to head, Rachel borrows $7 from Monica's purse and then puts the purse down on the table. She puts it down on her left hand side but it then switches to her right hand side without her touching it and then back again. (00:22:25)
The One with the Stoned Guy - S1-E15
Continuity mistake: After Chandler walks in on Ross and Joey's dirty talking, Chandler walks towards the door and Joey turns to Ross. In the next shot Joey has turned around and is facing Chandler. (00:13:15)
The One with Two Parts: Part 1 - S1-E16
Continuity mistake: When Ross, Carol and Susan are watching the birth video, Ross has his hand on the fake baby's head. In the next shot his hand is on its shoulder. (00:15:50)
The One with the Stoned Guy - S1-E15
Continuity mistake: When the monkey is hanging off Ross's date's hair, in one shot he is hanging upside down, then in the next he is hanging the right way up. (00:06:35)
The One with the Candy Hearts - S1-E14
Continuity mistake: In the restaurant when Ross asks Carol to join him, his date has her hand with the chopsticks in it up on the table. In the next shot it is gone. (00:14:55)
The One with Mrs. Bing - S1-E11
Continuity mistake: When Joey shows Ross the photo of his parents wedding, Ross is holding a magazine. In the next shot, the magazine is gone. (00:15:00)
The One Where the Monkey Gets Away - S1-E19
Continuity mistake: At the end of the episode when Monica say high school wasn't her favourite time, Joey is stood behind the sofa next to Chandler. In the next shot, he has moved to the other side of Phoebe. (00:21:15)
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 with the Thumb - S1-E3
Continuity mistake: When Ross starts talking about his dog at the start of the episode, Monica has her left foot on the floor and she is holding onto her right leg. As Ross says 'send' both of her feet are on the sofa and her hand is between her legs. (00:01:00)
The One with the Ick Factor - S1-E22
Visible crew/equipment: When Rachel is telling everybody about the dream she had with her, Chandler and Joey in it, Ross stands up and starts talking. When he does, the stage lights are visible in the TV. (00:13:40)
The One with the Thumb - S1-E3
Continuity mistake: After Chandler shouts at everyone because they are complaining about his smoking, there is an awkward silence and we see a shot of Rachel leaning on the sofa with her arms spread. In the next shot she isn't leaning on the sofa anymore. (00:14:25)
The One with the Stoned Guy - S1-E15
Continuity mistake: When Phoebe asks Chandler if he wants to be a chef, Monica says 'Hi, there' with her right hand raised. In the next shot, it is on her lap. (00:03:05)
The One with Two Parts: Part 1 - S1-E16
Visible crew/equipment: In Central Perk when Ross realises he's going to be a father, a black and white boom mic can be seen at the top of the screen. (00:16:50)
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 Where Nana Dies Twice - S1-E8
Continuity mistake: When Joey shows Chandler the TV in his pocket, at first he takes in out and is holding it in one hand. In the next shot, he is holding it in both hands. (00:14:00)