About 30 minutes in, when Monk was looking for the piece of red fabric, the alarm clock first showed 4:01. After one camera change it showed 4:04. Then another camera change, it showed 4:01 again.
When Lancaster claims the "chair test" was a joke, he waves his hand dismissively. In the next shot, this same hand is suddenly busily writing on the pad.
When Monk begins his self portrait, he starts in the approximate middle of the paper. When the camera angle changes he is suddenly drawing up from closer to the bottom.
When Sharona talks to Chet, her hands change position between shots a few times.
Mental health facilities such as the one in this episode are careful to keep all doors and windows leading outside carefully locked to prevent patients from wandering off. The second-floor window of Manny's room would certainly have been locked to prevent patients from doing exactly what he did - climb out onto the roof.
[16:35] When Manny tries to take a photo of Santa, he is using the flash through a closed window. But all he'd receive is a picture of the flashlight mirrored by the window pane.
When Mr. Monk checks the back of Trudy's picture, the secret message from Sharona is visible even though it was facing the wall and wasn't exposed to light.
As they are reenacting the murder, the text in the newspaper article about the murdered doctor is shown in close up and revealed to be dummy text.
When Dr Lancaster is on the roof dressed as Santa Claus trying to retrieve the gun from the chimney, Monk goes out on the roof with Manny. There is a harness attached to Monk as he helps Manny from falling off the roof.
When Dr. Lancaster is first seen, he very clearly says he's prescribing Neurotriptyline for Monk. There is no such drug. Most likely he is mispronouncing Nortriptyline.
When Manny is showing Monk around his room, there is a map of Antarctica on the wall labeled "North Pole". Antarctica is at the South Pole.