Other mistake: In the scene where Mr.Babcock is murdering the woman in the cinema, we see the woman has a big bucket of popcorn. As he starts strangling her, only a small amount of popcorn falls out. When the camera turns toward the floor there is a lot more popcorn on the floor and it it is spread right across the screen.
Plot hole: Monk identifies the killer because of a shirt sleeve that gets ripped off... he noticed the button is sewed on in a unique manner with the threads parallel as opposed to the "standardized" crisscross pattern. The killer and Monk coincidentally use the same dry cleaner and Mrs. Ling sews buttons parallel because it's her style. Dialogue in the episode indicated she is the only one who sews buttons on like that. But I checked over 20 button up shirts here at my house and the ratio of crisscross to parallel button sewing methods is 50% of each - hardly some unique identification method.