The Fuzzy Boots Corollary - S1-E3
Continuity mistake: When Sheldon is telling Leonard that he didn't ask Penny out, there is a yellow food-bag on the counter between them that keeps changing its position with every shot change. (00:11:15)
The Hamburger Postulate - S1-E5
Continuity mistake: At the end when Sheldon has got a burger, the amount of burger left changes depending on the camera angle. Most noticeable when he asks Penny about permanently reserving the table - between shots the lettuce disappears, the bun changes colour, and the bite marks change.
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)
The Big Bran Hypothesis - S1-E2
Continuity mistake: At the end of the episode, when the gang is reading the furniture assembly instructions, a pencil in the cup in front of them suddenly vanishes. (00:19:00)
The Fuzzy Boots Corollary - S1-E3
Continuity mistake: When Sheldon, Howard and Raj play Jenga, there is one block sitting at the top of the tower. When Leonard walks in, there are now two blocks at the top, before we actually see Howard place the second one on the top. (00:08:10)
The Luminous Fish Effect - S1-E4
Continuity mistake: When Leonard and Sheldon are walking downstairs, Sheldon stops and pins back up the caution tape that is in front of the elevator. Between shots, the other strip of caution tape untwists itself. (00:01:25)
The Fuzzy Boots Corollary - S1-E3
Revealing mistake: When everybody is on their laptops at the beginning of the episode, you can clearly see in many shots that their screens are black. (00:01:20)
The Luminous Fish Effect - S1-E4
Continuity mistake: When Sheldon's mom is saying the prayer, the bread roll that Raj is holding swaps hands, and then it disappears in the shot where she tells them they don't have to join in with the end of the prayer.
The Big Bran Hypothesis - S1-E2
Visible crew/equipment: When Sheldon is mentioning about the coffee table having a garage sale, there is a shot of Leonard with all the stagelights reflected in the kettle. (00:06:40)
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)
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 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)
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.
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 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)
Chosen answer: It does. The variable cleanliness of the cab and the habit of taxi drivers to engage customers in small talk would be too much for Sheldon to handle, and not an option for him.
Captain Defenestrator