The Grasshopper Experiment - S1-E8
Continuity mistake: When Raj's parents tell him to put his laptop down and gather his friends, they all huddle around the one-seat sofa. Sheldon is crouched to the left of the seat with his hands on the sidearm, yet in shots angled at the left he has disappeared from his place (as to not obstruct these particular angled shots.) (00:02:00)
The Grasshopper Experiment - S1-E8
Continuity mistake: When Raj is drinking with Lalita in the bar Penny works at, in one shot he raises his Grasshopper drink to his lips with his left hand. In the next shot it's in his right (this is just before he says "Oh my God, you've lost so much weight").
The Grasshopper Experiment - S1-E8
Continuity mistake: When Raj starts talking to Penny after he drinks the Grasshopper drink in her apartment, in one frontal shot he lifts the drink and begins to drink but in the next shot from behind the drink is back on the counter.
The Cooper-Hofstadter Polarization - S1-E9
Continuity mistake: The video of the fight we see on Youtube is different to what we saw earlier in the episode.
Continuity mistake: When Sheldon is talking with Leonard in the kitchen at night, as he pours himself a mug of milk the mug moves between shots. (00:04:50)
Continuity mistake: Sheldon puts the bag in the bin at the start of the episode so that the handles are facing the ground. When he gets the bag out, the handles are at the top of the bin.
Continuity mistake: When Penny is telling Howard and Raj about Sheldon's "drug addicted cousin" her scarf keeps jumping in and out of her jacket.
Continuity mistake: In nearly every episode, Sheldon demonstrates his OCD when he knocks three times on a door, says a person's name, and repeats that sequence twice, for a total of three iterations, but in this episode he radically departs from that behavior not once but twice. First, wanting to talk to Leonard, he knocks four times, says "Leonard" and repeats the sequence three times for a total of four iterations. Later, wanting to talk to Penny, he knocks four times and says her name, repeating that sequence at least four more times. Given that this was the first season, I assume the writers had not yet settled upon the 3x3 routine as a comic behavior.
Continuity mistake: During the whole opening scene, the way in which Leonard holds the letters in his hands keeps differing.
Continuity mistake: Sheldon puts the bag of takeout into the lobby trash can and as he does so, the whole bag turns sideways. At the end of the scene, he takes the bag out of the can and it's upright without him turning it.
The Pancake Batter Anomaly - S1-E11
Continuity mistake: After Leonard gives Penny her pile of mail, she alternates from holding the pile by her side to at her front between each shot. (00:01:10)
The Pancake Batter Anomaly - S1-E11
Continuity mistake: When Sheldon walks out his bedroom wrapped in a comforter, he drops it on the higher level before the drop down to the kitchen. In the next shot as he walks away from the fridge, half of the comforter is now spread over the lower level without anyone moving it. (00:05:10)
The Pancake Batter Anomaly - S1-E11
Continuity mistake: The way Penny holds her notepad when talking to Sheldon at the Cheesecake factory keeps changing instantly. (00:10:10)
The Bat Jar Conjecture - S1-E13
Continuity mistake: When Raj is talking about psychological warfare he has a fork in his hand in one shot and not in the next. (00:08:40)
The Bat Jar Conjecture - S1-E13
Continuity mistake: At the end, when Leonard is gloating over beating Sheldon, his position on the sofa keeps changing in relation to the arm. (00:18:50)
The Nerdvana Annihilation - S1-E14
Continuity mistake: In the scene where Leonard is packing up his collectables to sell, at 15:32 he is dusting what appears to be a Hasbro Marvel Legends Yellow jacket figure, which also has a yellow post-it on the back. At 15:42 he is still dusting a figure but he now holds a different (darker but not yellow) figure from the same series, with no post-it on the back, and the Yellow jacket figure is now on the table (house left of the box of collectables). At 15:56 The Yellow jacket figure has disappeared from the table, and it is not the figure in Leonard's hands either. (00:15:30)
The Pork Chop Indeterminacy - S1-E15
Continuity mistake: When Sheldon says that his sister was sent to spy on him, his arms are behind his back. Camera cuts and they are now at his side.
Continuity mistake: When Penny asks Sheldon "What trauma?" in the electronics store, Sheldon holds two boxes - one with an orange side on top of one with a black side. In the next shot the black sided box is on top of the orange one. The boxes then continue to swap places during the whole scene. (00:11:50)
Continuity mistake: In the electronics store, when the man asks Sheldon which hard drive he should buy in the first shot he is holding the green box in his left hand. Next shot it is under his arm. (00:12:50)
Continuity mistake: Penny throws a black iPod out the window, but when Raj comes in having found it on the ground outside he has a silver one. (00:01:00 - 00:02:45)