Mr. Monk Gets Cabin Fever - S3-E12
Continuity mistake: When the guys are following Disher, after he passes the store, the driver side window is rolled up. But in the next shot, it's rolled down.
Mr. Monk vs. the Cobra - S3-E11
Character mistake: Monk says that Croatia is a new country and didn't exist until 1992, but Croatia gained independence in 1991. However, Croatia still existed prior to that, it was just part of the Socialist Federal Republic of Yugoslavia. So things could still be made in Croatia.
Mr. Monk and the Red Herring - S3-E10
Continuity mistake: In the pet store, the bird's cage has a plastic covering on the sides. But in various shots, either one or both coverings are missing.
Suggested correction: This is wrong, it was just the way the camera angle made the cage appear where I couldn't differentiate the space between the bars, so it looked like there must have been a covering.
Mr. Monk and the Game Show - S3-E8
Continuity mistake: The way Lizzy is holding the envelope changes in the different shots. Plus, the flap is down at first but then up later.
Mr. Monk and the Girl Who Cried Wolf - S3-E6
Continuity mistake: At the end, when Sharona and Monk are leaving, Sharona opens the car door. But in the next shot, it's closed.
Mr. Monk and the Girl Who Cried Wolf - S3-E6
Continuity mistake: When Sharona sets the two grocery bags on the counter, the position of the items change after they tip over and after she picks the one on the floor up when talking to her ex.
Mr. Monk and the Girl Who Cried Wolf - S3-E6
Continuity mistake: After Sharona's class, she's holding her book upside down when talking to the teacher (so the binding can be seen). In the next shot, the book is held right side up. Afterwards, when she's walking away, the book is upside down again.
Continuity mistake: When the commissioner is getting into the car, he opens the door pretty wide. In the next shot when his hat gets taken, he's seen opening the door again.
Mr. Monk and the Blackout - S3-E3
Continuity mistake: When Sharona is trying to comfort Monk after he gets off the elevator, she's holding his hand and rubbing his arm. When Michelle walks away, in the next shot, Sharona's hands are on Monk's chest and shoulder.
Mr. Monk and the Blackout - S3-E3
Continuity mistake: When the second bomb is being placed (after the restaurant scene), the bomber is holding the bomb by the ends (the short side). In the next shot it's being held by the front (the long end).
Mr. Monk and the Panic Room - S3-E2
Continuity mistake: When Monk gets a glass out of the cupboard, he closes the door. When he picks up the phone, the door is open.
Mr. Monk and the Panic Room - S3-E2
Continuity mistake: When Monk and the Captain are in the recording studio room, the candles next to the framed pictures are lit. When Monk walks past them, they're unlit. They're lit again in the next shot.

Mr. Monk and the Girl Who Cried Wolf - S3-E6
Continuity mistake: When Meredith's husband has his heart attack, his head falls into the soup bowl. However, in one shot the bowl (with his head in it) is tilted; after a cut, it is suddenly flat on the plate.
Mr. Monk and the Election - S3-E15
Character mistake: Captain Stottlemeyer uses gloves to pick up the letter found sticking out of the pipe. He doesn't actually put the glove on, but still uses it to avoid getting any of his prints, skin, etc on the paper. But then he uses his other, non-gloved hand to open up the note and hold it, and Monk also touches the letter. If he cared about not contaminating evidence by using the glove, he wouldn't touch it bare handed and if he didn't care about touching the note bare handed, then he wouldn't have used the gloves.

Mr. Monk Takes Manhattan - S3-E1
Continuity mistake: The angle of one of the victims' heads visibly changes from shot to shot while Monk is investigating the scene. (00:05:30 - 00:06:25)
Mr. Monk Gets Stuck in Traffic - S3-E13
Continuity mistake: When Monk is talking to Daniel on the radio, in the shots from inside the truck, Krystal is standing next to the open door, to Monk's left when he turns to face her. In the shots from outside the truck, Krystal is now away from the door and to Monk's right.
Mr. Monk Gets Stuck in Traffic - S3-E13
Revealing mistake: When we see a shot of Steve in his car while in the dump truck, it's shot in a studio and the shot is reused each time we see Steve in the car. The shadow is completely different and at one point, when they show Steve in the car with the car and truck bouncing around, you can tell the image is just shaken, rather than the dump truck actually bouncing. There are loose chains hanging off the inside walls that don't move or sway at all.
Plot hole: In this episode the police are using the toupee of the commissioner to identify the torso as the missing woman. In reality that would not be possible since cut hair is missing the root and therefore only contains mitochondrial DNA, which can only be used to identify a human's relation via the female lineage, as it is the same in relatives and passed through generations from the mother to her children. Furthermore, the identification of said torso would likely not be sufficient proof that Mr. Harley killed her as he may claim that somebody misled him to believe she was in Paris by placing the telephone call he mentioned to police.
Mr. Monk Gets Cabin Fever - S3-E12
Continuity mistake: When Natalie opens the captain's smashed cake in the van it is vanilla, but when she gives it to the captain he grabs some with his fingers to eat and it's chocolate with only a bit of vanilla. (00:28:40)
Mr. Monk Takes Manhattan - S3-E1
Continuity mistake: There is a shot where Randy is trying to cross the police tape and an officer stops him, and he can be seen briefly arguing. However, when it cuts to the next shot, we see that argument happening again in the background, repeated. (00:07:20)