Cop shows are unusual amidst the onslaught of fantasy, sci-fi, and magical girl shows seen in all corners of the anime spectrum. Although certain shows like Cowboy Bebop have elements, it's hard to find a real-world action show that has any merit. One show worth your attention is City Hunter, a program so popular it even spawned a live-action version featuring Jackie Chan. The exploits of Ryo Saeba and his investigative firm bring back memories of the best days of Magnum P.I., even if said detective is a little too lecherous for his own good.
To try and go into detail about this show's plot would be hard, as it spans no less than two television series, several feature films (animated and live action), and at least a few specials. It's also episodic, so although there are a few continuing plotlines, the stories are pretty self-contained. Ultimately, the show is about Ryo Saeba, a P.I. who's a sucker for big cases involving pretty ladies. A bit of a pervert when he thinks nobody's watching, when he is around gorgeous women he becomes the definition of cool. For example, in the first episode, to save a damsel in distress, he shoots through his own hand to take down a killer. Ever nonchalant, he doesn't even flinch--until the girl is out of the room, at which point he starts freaking out! Despite Ryo's wandering eye, he's a good guy with plenty of style to spare and a desire to do what's right--particularly if it impresses somebody of the opposite sex.
Most of City Hunter's run was in the 80s, and as such it looks like anime from that time period. Thankfully, at that point in time television anime still had a budget, and so it often looks better than its modern counterparts that cover up a lack of cels with bright colors. However, depending on the director, animation team, and such, this fluctuates greatly. Nevertheless, one should also investigate the manga, which has detailed and exciting artwork that just can't quite always be animated well.
What makes City Hunter a good show is that it tends to be fun more often than not, and during the second series (City Hunter 2) things pick up a great deal overall. There's not a lot of character growth, but for a "realistic" action-comedy show, that's not always necessary. There are plenty of A grade episodes to be seen--however, overall it's a B because the show can be quite scattershot. Nevertheless, it's worth investigating.
City Hunter -- violence, profanity, brief nudity (on rare occasions) -- B