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.
Factual error: Sheldon says the movie Gremlins "baffles" him in regards to the instructions being very clear."Don't feed the gremlins after midnight." However, this instruction is only for caring of a Mogwai, not gremlins (the gremlins are formed by Mogwais eating after midnight). Sheldon, of all characters, would not mistake Mogwais for Gremlins and he would know the exact wording of the instructions.
Factual error: Sheldon's voice becomes squeaky when helium is pumped into his office. But if the room contained enough helium to make his voice squeaky just by breathing, Sheldon would actually be suffocating due to lack of oxygen, the lighter helium having displaced the heavier oxygen. You can do it with a balloon because you can breathe normal air between huffs, but if you're in a room full of helium, you won't last long before passing out and asphyxiating.
Character mistake: When Howard finds the stack of paychecks in Sheldon's drawer and wonders why Sheldon hasn't cashed them, Sheldon explains that he's saving them to purchase items that have yet to be invented and he doesn't trust banks. But just keeping them in his drawer doesn't do anything, because checks are voided after a set period of time and they would eventually become worthless.
Factual error: In order to get vengeance on Kripke, Sheldon mixes a solution of hydrogen peroxide, saturated potassium iodide and liquid soap, creating a large foam blob. This is easy to replicate, and what you will get (and look carefully, it is what Sheldon gets) is a huge, aerated foam mass consisting almost completely of bubbles. It is barely heavier than air, and if dropped from a ceiling as we see later in the episode it would float harmlessly about the place. Whatever it is that drops on Kripke and his visitors later, it isn't the foam mixture we see earlier - it looks like some sort of custard mix.
Character mistake: When Howard says they should put the time machine into the elevator in 1-14 "The Nerdvana Annihilation", Leonard states the elevator had been broken for two years, but in this episode, we found out the elevator broke seven years earlier. (00:03:10)
Character mistake: When Sheldon is placing his order near the beginning of the episode, he states "In the mid-18th century, King Rama IV of Siam..." King Rama IV was born 18th October 1804 and died 1st October 1868. His reign was during the mid-19th century (1851-1868).
Continuity mistake: Penny asks Sheldon if the roommate agreement contains something about girlfriends, which Sheldon dismisses as unrealistic. However, it was revealed in season 2 that there is a cohabitation rider. This was revealed when Sheldon asserted that Leonard's girlfriend, Stephanie, was officially living with them.
Factual error: At the very end of the show, Leonard and Raj arrive at the hotel, which is supposed to be in Geneva, Switzerland, to visit CERN. If you look out the "hotel" window, it looks like they're half way up Matterhorn, but in Geneva there are no big/rocky mountains anywhere near the city itself.
Character mistake: Sheldon refers to using the spray bottle as negative reinforcement when it is in fact positive punishment. Reinforcement involves encouraging a behaviour by either adding or removing a stimulus whereas punishment involves discouraging a behaviour by adding or removing a stimulus. Sheldon would know the difference.
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).
Continuity mistake: When Howard is proposing to Bernadette she is holding a tray with two glasses on it. The glasses are next to each other, side by side for two shots of her. Then in the third shot the glasses move so one is behind the other. Then in the fourth shot they are side by side again.
Continuity mistake: When Penny is making French toast and Sheldon walks into shot, the edges of the toast are not covered in egg whilst the rest of the toast is. About 30 seconds later, Penny asks Sheldon if he wants French toast and the edges are still not covered in egg. The shot cuts to Sheldon. In this shot we can see Penny's hands to the far edge of screen and fully raised showing us that she hasn't dipped the edges of the toast in egg yet. But in the next shot, we can see the whole of the toast is covered in egg, including the edges of the toast that were clear a moment earlier. (00:00:20 - 00:00:50)
Continuity mistake: When Raj and Leonard are sitting on the couch, Leonard is holding Watchmen and Lost in Space in his left hand and Young Sherlock Holmes in his right. In the very next shot when Leonard turns his head after Raj speaks, Watchmen has disappeared.
Factual error: Maybe I'm wrong but I've had the dislocated shoulder experience and shouldn't hers be in a sling at her side not across her front right to left? This just seems odd.