Character mistake: In his conversation with Missy Sheldon makes it clear that his superior intelligence is a result of a random, mutated gene. Since Missy isn't similarly intelligent she obviously isn't carrying this mutated gene (which would be a billion to one shot anyway) so her offspring wouldn't inherit it. Sheldon would know this - his offspring would carry the mutated gene for superior intelligence, Missy's would not. Anyone knowing enough about genetics to use the term 'randomly mutated gene' understands enough to know that the mutated gene would only be expressed in a direct line from the carrier - Sheldon. Also bear in mind he has a model of the DNA molecule in his living room - it is obviously an interest.
Plot hole: In Series 1, Episode 7, "The Dumpling Paradox, " Sheldon makes an oblique reference to his financial status: "Frankly, if I could afford the rent, I'd ask you [Leonard] to leave, " meaning he cannot pay the rent on his two bedroom apartment by himself - not that he doesn't want to, he can't. However, in "The Execlsior Acquisition, " we find that he does not even cash his pay cheques. He doesn't even deposit them into a bank account - he leaves them in a drawer in his desk. In Series 2 Episode 14 "The Financial Permeability", he lends Penny a large amount of money from a huge bankroll he just happens to have lying about - again, without cashing his pay cheques! He is obviously independently wealthy. Either he doesn't have enough money to afford the rent or he has enough to work without being paid while practically giving large amounts of cash away. Can't be both.
Character mistake: Sheldon is wrong about the physics of pushing the furniture up the stairs. This is not just a question of work (energy) but also of power (energy/time). Since he and Leonard cannot produce an unlimited amount of energy per second, they may need to push it slowly, contrary to what he asserts. This is a very serious and very basic error for Sheldon considering he has a PhD in Physics.
Continuity mistake: In episode 16 of season 1, Penny is talking about making a cake for Leonard's birthday and states she knows his birthday because she was checking a horoscope for him. But in this episode she does not know when his birthday is, which leads to Leonard revealing he was secretly reading Penny's diary. (00:40:00)
Other mistake: In the scene where Leonard threatens to open his production era, we see the Star Trek The Next Generation Geordi LaForge (w/o visor) action figure. The uniform worn by the action figure is only worn by Geordi in Star Trek Generations the movie, not the television series.
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.
Character mistake: Sheldon says (during Fun With Flags) "But like the flag over Fort Sumter, I'm still here." The Sumter flag was lowered when the fort surrendered to the Confederates. The flag over Fort McHenry was never taken down despite the British bombardment and inspired Francis Scott Key to write the National Anthem about it. Whouldn't a vexologist know that?
Other mistake: When Mary is rubbing VapoRub on Sheldon's chest, she makes a comment that last time she saw that there was no hair, to which Sheldon replies "Yeah it filled in last year." In Season 1, Episode 11, Penny rubs VapoRub on Sheldon's chest and a very clear remark about chest hair was made. Season 1 for the show was 4 years ago.
Continuity mistake: In "The Luminous Fish Effect" (S1E04) Sheldon tells Penny he weighs 140 pounds. In "The Porkchop Indeterminacy" (S1E16), he tells his sister he weighs 165. 25lbs difference would be a noticeable change, but he looks the same.
Continuity mistake: Sheldon and Amy are talking while drinking tea together. The entire time, Sheldon has his hand next to his other hand, but in an individual shot, his hand is suddenly on his other wrist.
Factual error: Characters sometimes talk to one another on video calls, and a character (A) often watches these from an angle rather than face on, allowing the viewer to see both character A and the character on the call (B). When this happens, character B is shown turned a little to one side, to give the impression that they are looking directly at character A (as they would be if they were present in the same room). In reality, this means that character B is looking off to one side of their monitor and not at the image of character A, something that it makes no sense to do. No matter what angle the laptop is shown from, provided character B is visible, they should appear to be facing the viewer.
Factual error: The foam Sheldon used on Kripke consists of mixing ordinary dish soap, hydrogen peroxide, and saturated potassium iodide. That kind of exothermic reaction is dangerously flammable, so it should have severely burned Kripke and the members of the university board when it fell on them.
Character mistake: While bedridden after his vasectomy, Howard asks Bernadette to grill up a couple steaks instead of picking up Chinese for dinner. She gives him a disapproving look and he says "Kung Pao Chicken, please." A major ingredient of Kung Pao Chicken is peanuts. A major allergy for Howard is peanuts. (00:09:04)
Continuity mistake: In the early episodes, Leonard is depressed and buys cat stuff. He comments that he has taken in to account Sheldon's asthma with a hyper allergenic cat. Later in the series it is Leonard with asthma and Sheldon who gets all the cats.
Character mistake: While driving towards the restaurant, Amy mentions she uses dandruff shampoo for her "dry" scalp. Dandruff and dry scalp have the same main symptoms, which are falling flakes and an itchy scalp, but they are two different conditions. In dry scalp, the skin gets irritated and flakes off. With dandruff, the cause is too much oil on the scalp. That excess oil causes skin cells to build up and then shed. Therefore dandruff shampoo should be used for an oily scalp, not a dry scalp. (00:11:19)