The Hawking Excitation - S5-E21
Character mistake: The sign on Howard's laboratory door reads "RESTICTED area".
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 Vacation Solution - S5-E16
Character mistake: In Amy's lab, when she is complaining to Sheldon how she gave him the simplest tasks, and that they're still not done, Sheldon replies with how Einstein failed math because he wasn't challenged, and that it is the same case with him. Einstein never failed math. Sheldon of all characters should know that.
The Shiny Trinket Maneuver - S5-E12
Character mistake: The bird that Sheldon is afraid of is referred to in the scene as a "Blue Jay." In fact, the bird is a Black-throated Magpie. A Blue Jay is a little smaller, has a shorter tail, does not have the darker plumage, and can be quite obnoxiously vocal.
The Russian Rocket Reaction - S5-E5
Character mistake: Sheldon says he drove for hours in a bus to meet Wil Wheaton. He meant he rode on a bus. Sheldon never would've made that semantic error.
The Werewolf Transformation - S5-E18
Character mistake: In S5-15, Sheldon is asking his friends to take him to SuperCuts. But in this episode, he goes to Mr. D'Onofrio's barber shop where he says he's gone to since he moved to CA, and he said D'Onofrio had his "haircut records."
The Friendship Contraction - S5-E15
Character mistake: Sheldon told Leonard at lunch that they were no longer friends, only roommates but later that day, he told start moved up to ninth place on his friends list because he dropped someone (can't remember who) off his friends list). In actuality, it would've bumped him up to eighth place because he kicked Leonard off the list.
The Speckerman Recurrence - S5-E11
Character mistake: Throughout the show, Sheldon denounced his mother's religion and God. To the extreme. In this episode, he said, "More sense than God gave a mouse." He would never invoke God for any reason.
