Character mistake: Sheldon says he once borrowed his sister's makeup and ended up with pink eye but he was dressed as a zombie and he won second place. There is absolutely no way his hyper-religious mother would let him dress as a zombie.
Continuity mistake: When Howard is almost done moving out of his mother's house, there is a table in front of him with a measuring device in the top corner. In one shot, the device moves to the opposite corner of the table. In a few shots later, the item returns to its original position.
Revealing mistake: Throughout the series much is made of the fact that Howard's mother is grossly obese, and in The Engagement Reaction (S4 E23) it is even mentioned that her arms are hugely oversized. However, at the end of this episode we see her arms when she grabs Raj to stop him climbing out of the window, and they are the arms of an average-weight woman - they aren't overweight at all.
Continuity mistake: In this episode Penny mentions she has a sister ("I remember my first bikini wax. My sister did it with duct tape and melted Crayolas"), but in future episodes she has only a brother.
Continuity mistake: Sheldon and Amy are talking while drinking tea together. The entire time, Sheldon has his hand next to his other hand, but in an individual shot, his hand is suddenly on his other wrist.
Continuity mistake: Sheldon is taking composition books out of a box that was sent to him. When he first starts pulling them out, he has two composition book in one hand. Two shots later, he has only one composition book in his hand, but he is picking up a second one.