Continuity mistake: In the early episodes, Leonard is depressed and buys cat stuff. He comments that he has taken in to account Sheldon's asthma with a hyper allergenic cat. Later in the series it is Leonard with asthma and Sheldon who gets all the cats.
Suggested correction: Presumably Sheldon also buys non-allergenic cats.
Continuity mistake: In the beginning of the series Sheldon knocks on Penny's door only once. Later in the series Sheldon explains he has always knocked 3 times because he walked in on his father with another woman.
Continuity mistake: Throughout the early episodes, shots of Sheldon's bathroom shows the door open and a rack of towels against the wall. Later episodes show that that area is actually where the sink and mirror are.