Continuity mistake: When the murderer strangles his victim, her music box is dropped on the floor several feet away. Though no one has touched it, the music box is somehow lying only a few inches from her body when the police arrive to investigate. (00:01:20 - 00:04:40)
Revealing mistake: During the fight in the kitchen, piles of plates are knocked to the floor. The plates that we see hitting the ground all remain intact. Nevertheless, we hear the sound of plates smashing, and when the aftermath is shown, there is broken crockery everywhere. Clearly the plates used were made of plastic or some other unbreakable substance to minimise injury to the actors and stuntmen.
Continuity mistake: Monk leaves his front door standing open in anticipation of his new friend's arrival. No one goes near the front hall, but when Hal gets there, he lets himself in by opening a door that has somehow closed itself. (00:10:30 - 00:11:50)
Continuity mistake: The position of Gail's body moves at the crime scene from when officers cover her head with the tarp to when the Captain uncovers her a few moments later. At first, her left hand is under her forehead and covered by her hair, but then the hand is away from her head and not covered by hair.
Continuity mistake: Monk pushes aside a glass of milk on the table in his apartment. It is mentioned multiple times during the series that Monk is afraid of milk. He would never have any in his home, and certainly wouldn't have a neutral reaction to it. Also, the milk appears to have spilled a little as it was pushed, and yet in the next shot there is no milk on the table. (00:33:00)