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.
The Big Bran Hypothesis - S1-E2
Character mistake: Sheldon does not do his 3x OCD knock on Penny's door.
Suggested correction: That quirk wasn't brought into the series until halfway through the first season. The one-knock Sheldon happened a couple of times before it became the three knock habit.