Continuity mistake: Jerry's apartment in the first episode was on the fourth floor. Then in the next bunch of episodes it was 3A. It then changes to 5A for the remainder of the series.
The Stakeout - S1-E1
Continuity mistake: In the video store, we see Elaine put back Cocoon 2 and then doesn't pick up another video. But when she takes the video from Jerry, she's holding a video in her other hand.
The Stakeout - S1-E1
Continuity mistake: At the video store, when Jerry shows Elaine the adult video, he holds it with both hands with his right hand fingers under the box. In the next shot, he's holding it with only one hand, with his right hand holding it by the side.
Continuity mistake: Up in Vermont, Jerry is beginning to have a rather tense conversation with Vanessa. In the close-up after the watch comment (when he is thinking of the delightful drive home ahead) Seinfeld is all smiles, but in the shot before and after he is staring with a tense expression. (00:17:20)
Continuity mistake: Elaine comes out of the bathroom sniffling and with a big stripe of toilet paper. She rubs her left side of the nose when she asks "Didn't work, huh?" In the next shot with George answering with a burst of nervous laughter, Elaine's hand is in a different position, and George himself is clutching a tissue to his chest. (00:16:25)
Continuity mistake: Kramer runs off to meet the girl he just spotted through the binoculars, and George walks through the door and crashes on The Couch. The pillow is straight-ish, very angled and straight again during the conversation, with George's hand changing position between shots. (00:15:45)
Continuity mistake: The scene with Elaine on The Couch plotting the demise of the cats begins with a nasal spray on the kitchen counter. There's no trace of it after that initial shot, and it reappears only when Elaine comes closer for her cup of coffee. (00:13:00)
Continuity mistake: Jerry orders a tuna sandwich, and Elaine turns towards him and whines "Tuna?" Her hand is still on the menu, but when the angle changes her hand is on the neckline of her top. (00:04:50)
Continuity mistake: Jerry says that he wants to suggest to his current girlfriend, Vanessa, to go away for a couple days. George starts saying "No no no" and before the cut he raises his arm and lowers his head as to express concern. But in the new shot there's no trace of that change in posture. (00:02:45)
Continuity mistake: Elaine joins the other late at the table. She sighs and puts her sunglasses on the table, above an open menu. Brief cut to George, and the menu is on top of The Glasses. It is again at the bottom later. (00:01:45)
Continuity mistake: During the initial conversation, George keeps playing with the paper sleeve of a straw in certain shots, while his hand is in another position in the reverse shots (noticeable for instance when he says "Guys with cats...I don't know..." (00:02:30)
Continuity mistake: Jerry was forced to take the phone call thanks to Kramer. He is holding the receiver in a different way between shots - look at the position of the index, on the microphone in the second shot. (00:03:50)
Continuity mistake: In the hallway scene at the beginning, Jerry is holding a magazine. When the man comes out of the elevator, he is holding the magazine The Opposite way - easy to tell because of the Marlboro ad in the back, now flipped. (00:01:55)
The Robbery - S1-E2
Continuity mistake: Jerry is being grumpy on The Couch with his friends for giving up The Apartment, arms crossed. Notice his right sleeve, down till the wrist. When Carol enters the room with her husband for the first time, Jerry is still with his arms crossed but the sleeve is up, mid-length. (00:20:35)
Continuity mistake: When Elaine passes the newspaper to George, Jerry is sitting normally in the first shot, leaning with his right arm on the backseat in the next. (00:03:10)
The Robbery - S1-E2
Continuity mistake: Jerry is back from Minneapolis, and he puts The Keys down on the kitchen isle. When Elaine walks in with the plunger, The Keys have moved forward and more towards the center. (00:04:45)
The Robbery - S1-E2
Continuity mistake: During the initial conversation when the camera is on Elaine there's a chair prominently featured in the background, with a jacket on the back. The chair turns around between shots - in particular, when Jerry proudly proclaims himself "Master Packer" it is positioned at the desk instead of facing towards the camera. (00:01:10)
The Robbery - S1-E2
Continuity mistake: George is chatting with Elaine as he is waiting for Jerry. Elaine splits an Oreo cookie. When he mentions her "price range", she is not seen fully but her left hand is in frame and she brings it up. When George turns around to address Jerry though, it's her right hand to be raised. (00:03:20)
The Stakeout - S1-E1
Continuity mistake: Jerry and his mom are playing Scrabble; when she is picking the non-existent word, Jerry says that he's "nervous." Look at The Note between his hands. At the cut, the paper is in a different position and orientation. (00:17:45)
The Stakeout - S1-E1
Continuity mistake: When Jerry is making the "Lo Mein" joke inside the video store, he is waving his hands around. The hands are out of frame; still his right hand pops in view twice showing he is holding a tape - but he is holding none before and after and he positioned himself away from the shelf. (00:01:35)
Chosen answer: George does mention his brother in the episode "The Parking Space". They actually really never mention anything that contradicts the fact he had a brother though, it just isn't mentioned.