As I understand it, the law enforcement in Duskwall is very noir. As a result, its three branches fit into very specific stereotypes.
The Bluecoats are your typical corrupt beat cop. This means that they are the actual assholes tv shows cops as being; they beat store shop owners for "protection" money, they accept bribes to ignore criminals selling drugs or smuggling goods, they will "legally" kill people who either piss them off or piss off someone they have a debt with, etc.
The "nobility's lapdog" comes with the fact that they are still the "law enforcement" and as such, it's their job to arrest people. They just arrest whoever the nobility tells them to arrest rather than who the evidence tells them to arrest. In fact, the only time they would follow the evidence is if they haven't been bribed or ordered not to.
After that, you get the magistrates, who are your corrupt judges and as such weild the power to either save or ruin lives, depending on which way the blackmail and/or coin is flowing.
Then you get the Inspectors, which are your noir police detectives/private investigators. A case lands on their desk and they follow it through come hell or highwater. They can't be bribed or blackmailed by the criminal class and as such, they either finish the job or die in the line of duty.
Now to me personally, being unbribeable is vastly different from actually wanting justice. As law enforcement in Duskwall, they are still bound to obey the nobility. So to me, the inspectors are going to be a mix between Dirty Harries and the "nobility's enforcer", where they properly pursue a case, or they will pursue a case however they are told to do so.
Of course, separate and apart from everyone else, you get the Gray Coats, who are former Bluecoats that were almost killed by the nobility in attempt to cover up evidence via arson. So now they're gangsters who occasionally pretend to be Bluecoats, who basically live to piss off the nobility.
I could be wrong about any or all of this since I was trying to read up on it some of yesterday and most of last night.