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 Cooper Extraction - S7-E11
Continuity mistake: When Amy says that Penny never would have met Stan Lee, she draws a thick black line between "Penny" and "Stan Lee" on the board. The camera goes to Penny, and when it comes back to Amy, the line is blue and faint.
The Einstein Approximation - S3-E14
Revealing mistake: In the scene where Sheldon visits Leonard and Penny in bed, towards the end of the scene, Leonard turns off the light. As he does, if you look carefully, you can see that the light turns off before he hits the switch.
The Financial Permeability - S2-E14
Continuity mistake: When the guys are using the whiteboard to figure out their cinema plans, Sheldon draws two black X's on the board with a marker. In the following shots the appearance of the X's changes. (00:00:30)
The Bath Item Gift Hypothesis - S2-E11
Continuity mistake: When Sheldon emerges with all of the bath item gift baskets for Penny, he drops them on the floor and chair. Between shots one of the baskets on the chair changes from being an upright green-ribboned basket, to a toppled-over white-ribboned basket. (00:19:50)
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)
The Lizard-Spock Expansion - S2-E8
Continuity mistake: When the guys are in the Mars Rover test room, the time displayed on the large digital clock seen behind Howard jumps about inconsistently between each shot. (00:07:50)
The Hamburger Postulate - S1-E5
Audio problem: When Leonard and Leslie continue to practise together after the other two musicians have left (Leslie now being the only violinist), vibrato from a violin can be heard - from observing Leslie's left hand she is not applying it.
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 Einstein Approximation - S3-E14
Character mistake: When working in the Cheesecake Factory, Sheldon says "Bonne appétite," with a hard T at the end. He should know that this is incorrect and would be the first to remind others that the correct phrase is "Bon appétit" (pronounced "appéti" - it is masculine and the t at the end is not pronounced).
The White Asparagus Triangulation - S2-E9
Continuity mistake: When Sheldon is in Penny's apartment trying to get her to tell him why her relationship with Leonard failed, the yellow tee-shirt on the red pillow next to him keeps changing position.
The Hamburger Postulate - S1-E5
Character mistake: When Sheldon knocks on Penny's door to seek advice about the tie on Leonard's doorknob, he does not use his typical triple knock. (00:07:43)
First time he knocks 3 times is in Episode 2 of Season 1. First time knocking 3 times followed by saying the name is episode 10 of Season 1. The ritual of 3 knocks and 3 times saying the name and then stopping is Episode 5 of season 2.
Sheldon knocks three times because he once walked into his parents bedroom and saw his father with another woman. He's been doing the triple knock since he was a teenager.
Yet, he didn't in the show till Season 2. Funny ain't it?
Suggested correction: A one-time lapse in someone's usual behavior does not count as a "character mistake", even in the case of someone given to ritualistic behavior as Sheldon is.
Ah, but in an episode he explains he had developed this particular ritual at age 13 after walking into his parent's bedroom without knocking and seeing his father having sex with another woman. He says he started knocking 3 times since then and would never forget. So its not consistent and a mistake as this is not the only example one can name.
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)
Character mistake: Wolowitz warns his fellow halo players that their enemy is "charging his Plasma Rifle". This is impossible since only the plasma gun can be charged up. (00:20:00)
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 Benefactor Factor - S4-E15
Continuity mistake: When Sheldon and Leonard are in the kitchen discussing Leonard's upcoming night out with Mrs. Latham, there's a plastic bottle of orange juice on the table that keeps disappearing and reappearing as the camera switches between Sheldon and Leonard.
Suggested correction: The bottle is there every time, a few times because of the camera angle it is hidden by Sheldon.
The bottle of OJ is right next to a container of Coffee Mate, both items always visible when the bottle of OJ is present. When the OJ is not present, only the Coffee Mate is there, near enough the center of the table to not possibly be blocked out by either Sheldon nor Leonard.
No it is not right next to the coffee Mate, there is some distance with a phone in between. You can sometimes see the bottle behind Sheldon when he moves.
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 Barbarian Sublimation - S2-E3
Continuity mistake: When Penny sneaks into Sheldon's room at night to ask him about the online game, she leaves his bedroom door half open. But when she leaves, the door is shut. (00:07:35)
The Barbarian Sublimation - S2-E3
Continuity mistake: After Penny drops her bags of groceries whilst trying to unlock her door, the dropped items change their positions between shots (note the items against the wall.) (00:02:05)
Deliberate mistake: Very often, during Skype or other video chat conversations, the people on the computer screen look left or right when talking to someone who is not directly in front of the screen. Although it certainly helps the viewers understand who's talking to whom (which is why I assume it's a 'deliberate mistake'), it isn't realistic, since it implies that the people on the screen are actually constantly looking away from their computer instead of at their screen or webcam.