The Beta Test Initiation - S5-E14
Factual error: Penny picks up a hand gun, pulls back the slide to ensure the chamber is empty (which would discharge any round in the chamber too) and then releases it, leaving it closed. She then loads a magazine into the butt of the weapon and hands it to Leonard. He starts to smooch her and shoots himself in the foot. The gun could not have fired, because Penny would have to pull the slide all the way back and then release it again to load the first bullet. There couldn't have already been a bullet in the chamber, as the whole reason Penny pulled the slide back in the first place was to check just that.
The Beta Test Initiation - S5-E14
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?
Suggested correction: Sheldon is referring to the actual Fort Sumter flag, which still exists and is on display at the Fort Sumter museum.
Sheldon is not referring the actual flag (especially since if the actual flag is in a museum it's not still over Fort Sumter). Sheldon is referencing the line from The Star-Spangled Banner "our flag was still there." Key wrote The Star-Spangled Banner after seeing the flag over Fort McHenry was still there. The writers simply made a mistake and Sheldon the character would know which flag inspired the line. Incidentally, the actual Fort McHenry flag also still exists.