I see you edited the OP... Here's few more details.
I'd want a persistent crime system where things build up. And the PD to be a thing to fear; not inescapable or completely unavoidable, but something that will keep you on your toes and will not forget if you cross the law. That once you have enough on your account, they will vehemently hunt you down. And if it was you alone against the PD in open battle, there should be little chances for you to survive (no GTA style rampages agains an army). A system where petty crimes (if you are caught or are seen) get you fines but pile up for an eventual arrest warrant.
That could mean that you have your chances to bribe or speak your way out if getting caught, but if not succesful, you spend some jail time with additional fines. And for the player to care about that, there needs to be consequences and reactions from the game. Certain groups should take notice of these things and react in good or bad depending on the group, there should be dynamic quests that might "fail" or whose conditions might worsen while you are sitting your time, quests that are prevented by a criminal record, NPC's that do not interact with criminals. The criminal record effectively closing doors from you for certain quests, vendors, groups, buildings, locking you out of certain content. The PD's tolerance will get lower and you get hunted down more and more easily and need to be ever more careful. That's a lot of negatives, but there could also be positivie effects to it; such as certain NPC's only interacting with hefty criminal record, some gangs looking up to you more favorably when you are more like them, these interactions opening quests, interactions and merhcandise not otherwise obtainable.
There should be consequences, tradeoffs to think about, but since there is a lot of lawlessness in the city, there should also be opportunities that way.
There are skills for disguises and stealth to help you avoid the patrols and whatnot, there can be bribery of certain corrupt officials to ease your situation (though I don't think it should ever be possible to completely erase your record, nor should these ways of avoiding and evading ever be surefire ways to bail you out from responsibility, but just means of coping with it). The role you pick might have an effect - eg. if you are a cop, certain things might be looked past to preserve the reputation of the force and you might have easier time talking yourself out of a shit situation; if you are a corp, you might have easier and cheaper time bribing the officials; if you're a rockerboy, your celebrity status might be held as a mitigating circumstance for lesser crimes; if you are a solo, you are on your own with no bonuses....
You could keep track on your criminal record by hacking the police database (if you have the skill) from your pocket computer or what ever, or you could buy the info from "shady sources", or if you are a cop, you could have free access to it. That might sound a bit off the topic, but it could be considered in-game efficiency of the police to have that information under "guard" and have the player need to actually do something to obtain it (on regular basis if there should be updates).
...for examples sake.