In the city of San Francisco, there is no police rank of 'Detective'. The title is 'Inspector'.
The newspaper clippings in Monk's scrapbook (seen throughout the series,) which have headlines about Trudy's murder, are in fact articles describing the state of the economy.
The show intro shows Monk pouring boiling water on a toothbrush. The toothbrush is a normal toothbrush with a rectangular head. He then immediately brushes his teeth with a oval shaped multicolored toothbrush.
In the intro, Mr Monk walks out the door, then returns to adjust the position of an umbrella. Mr Monks O.C.D. would not have allowed him to hang it incorrectly in the first place.
Neither Sharona nor Natalie could possibly afford the houses they are shown living in given either woman's meager salary in San Francisco's notoriously high real estate market.
It is quite obvious to anyone who knows the real San Francisco Bay Area that the series was not filmed there. There are numerous references to extreme temperatures (when the Bay Area is famous for its year-round mild temps) as well as large rolling hills with nothing on them (SF is very densely populated with little left untouched).
Monk often undertakes actions which would result in evidence and investigation being compromised. Since he is no longer a police officer, these actions could result in cases being dropped or overturned upon appeal. There is no way that the San Francisco DA would allow Monk to perform the actions that he does.
Boxes on shelves above couch in Monk's apartment are often placed in different distances from each other. Monk with his urges wouldn't miss it and would arrange it correctly.
We see Monk constantly ask the detectives working with him (or the detectives themselves) to put out APB's (All Points Bulletins) or BOLOs (Be On the Look Out) to bring suspects and witnesses in. When found SFPD just walk up to people and place them in the squad car, taking them to the station for Monk and the detectives to interview even though it's pretty clear they didn't want to come. If there is no probable cause for an arrest or an active arrest warrant, the police can't make you go anywhere against your will. Once in a while, a wealthy or educated person will assert this and ask a lawyer to be present but this is rare.
Monk drinks bottled water, which isn't as pure as everyone thinks. It has about 200ppm (parts per million) of contaminants (compared to about 400ppm for tap water). Monk of all people would know that.