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.
The Rhinitis Revelation - S5-E6
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.
Suggested correction: Sheldon is probably joking when he tells Mary his chest hair filled in the previous year.
He doesn't understand sarcasm, hence, doesn't know when to be sarcastic.
And nearly all instances of Bazinga are not used in a funny circumstance. Indicating poor judgement and lack of knowledge regarding jokes.
They were funny to him. And to the audience.
The Launch Acceleration - S5-E23
Factual error: In all the episodes leading up to Howard and Bernadette's wedding, Howard was going into space to install a space toilet. But in this episode, Howard tells her they want his telescope, so he has to go.
Suggested correction: The space toilet was something Howard designed that NASA installed back in season 2. His mission to the ISS in season 5 was always about his telescope. In episode 5, "The Russian Rocket Reaction", Howard says NASA picked his team's design for the deep field space telescope that's going on the International Space Station in the spring.