While the cast is singing "For He's a Jolly Good Fellow," there is an edit after the first line. The result is that the first line is in a significantly different key from the rest of the song.
When everyone sings to Randy Disher at his surprise birthday party, strands of confetti from a party popper disappear and reappear in Natalie's hands several times between shots.
After Monk steps into the mud bath, "Officer Feelgood" tears off his clean shirt and tosses it to him. When Monk grabs it, it already has mud splattered on it (possibly from previous takes).
At the bachelorette party, after the stripper says, "We've had some complaints about you ladies...," the lady in the gold dress standing behind Teresa had previously been on the other side of the room as he was saying that. She "reappeared" behind Teresa after the camera angle changed.
In California, you can hold a suspect up to 48 hours before filing charges. There'd be no need for Stottlemeyer to rush Disher through the hotel in order to positively identify Theresa before she leaves. They could simply arrest her and still have plenty of time for Disher to identify her at the police station.
When Randy is hit by a car in Monterey, Stottlemeyer is in Monk's apartment in San Francisco. The next shot, Stottlemeyer and Monk are at the accident scene while the paramedics treat Randy. It is a 2.5 hour drive between the two locations and Randy would have been taken to the hospital by then.
[16:45] Lt. Bristo identifies himself as being from "Monterrey County Police." County law enforcement would be done by a Sheriff's office, not the police. Police departments handle law enforcement in municipalities.