Alucard from Hellsing. The guy's a cold-hearted bastard who loves to taunt his foes, but occasionally sides with the good as an Occult Detective.
Damien from Nightschool. He doesn't have stubble, but when you can stop bullets from reaching your skin, break bones by thinking about it, and can flatten NYC then you can be a nitpicker.
The Ur Example is, of course, John Constantine from Hellblazer of Vertigo Comics. He first appeared in Swamp Thing and spun-off into his own series some years later. (Swamp Thing showed the title character and John Constantine both active in the Crisis on Infinite Earths, although Constantine had only a very small role in the actual miniseries.) Constantine was created by Alan Moore based on his idea of "creating a character who looked like Sting", although Sting did not tend to wear trenchcoats or smoke (at least in public).
Though many writers have wanted to use Constantine in their series, DC denied them up until 2011 for fear of disturbing the "realism" of Hellblazer, even though Constantine (technically) inhabits the DC Universe, where a number of these Expies also reside. So, instead, various analogues and homages to John have appeared.
Also due to the above, Phil Foglio's update of Stanley and His Monster featured author Ambrose Bierce drawn to look like Constantine and playing much the same role. The series claims that his horror stories were based on truth, and he staged his own disappearance to avoid an Eldritch Abomination that was coming to complain about his depiction of it. People keep mistaking him for Constantine, who he calls a clown.
Neil Gaiman also created John Constantine's ancestor for The Sandman: Johanna Constantine. Even though she doesn't wear a trenchcoat, she's still a Magnificent Bastard. It's also implied that Constantines gravitate towards this trope especially if their initials are "JC".
Gambit from X-Men. Trenchcoat, stubble, mysterious past, dubious morality, all that's missing is the occult specialty.
Bigby Wolf from FABLES, who wears a trenchcoat and smokes thin cigars. The actual Big Bad Wolf, Bigby nonetheless has supernatural powers in him, and he's a detective too.
Beast is also one when he plays dress-up as Sam Spade or Philip Marlowe as he tracks down his demonic wife.
Pete Wisdom from Excalibur (even when he wasn't being written by Warren Ellis, his creator) was always like this. He's cleaned up a bit recently, though. Given up smoking and everything. Some fans have theorised that the original character was the Nineties Anti-Hero version of the character, and this is just a natural progression. Since Excalibur deals with Marvel's mutants in England, and lots of dimension-travelling, there's no question on the supernatural elements he's had to face.
Constance Johanssen was Warren Ellis's homage/parody of Constantine in the Pryde and Wisdom mini-series: "Constance Johanssen. Excellent occult detective. Has a habit of getting her friends killed. Two hundred at last count."
He was first appeared on earth in Britain (though he was actually born in Hell.) He considers himself American, though.
Samuel Burke from Spawn. Although overweight, the guy sees everything in the shades of grey. He smokes from time to time and when he taggles along with Spawn in his adventures, he becomes an Occult Detective.
Lt. Kellaway from The Mask qualifies. He wears a trench coat and smokes, even having a cynical yet-by-the-book look in his cases, and as he chases down the reality-warping maniac Big Head.
Harvey Bullock from the Batman mythos. This fat trenchcoat, cigar smoking, and donut eating cop is morally ambiguous in his cases. Lampshaded much in Gotham Central, which has the "freak beat"; the cops in that section deal with Batman's superpowered rogues gallery.
Marcos Bizancio from Bizancio, is an Argentinian equivalent.
While his literary counterpart below has some aspects of this, Anton in the Night Watch films is an even better example, wearing a trench coat and Cool Shades, and is clearly an alcoholic. Dealing with the "Others" is probably eough to turn anyone sour.
A recent addition to the brigade is Jimmy Stark, antihero of Richard Kadrey's Sandman Slim. Ragged & snarky, with a dark past and hellish powers; wearing a black silk trenchcoat with no irony, and smoking Maledictions, the cigarettes from Hell.
Fitz Kreiner, from the Eighth Doctor Adventures. Being Genre Savvy, he seems to know it, too. He smokes, wears a leather coat most of the time and a trench coat some of the time, swears more than basically anyone he knows, has Perma Stubble (because he's bad at shaving), and is a lower-middle-class Londoner and a GuileSidekick. He's also basically quite sweet and sensitive, but most characters, upon first meeting him, distrust him. Like all of the Doctor's compainions, he's given a Weirdness Magnet, to make sure when he wanders off, he discovers the nearest source of alien influence.
Simon R. Green's Nightside series has John Taylor, who is basically a nicer John Constantine.
Live Action TV
Castiel from Supernatural was influenced by Constantine's image. Though he's not British and doesn't smoke (as far as we know), he does wear a trenchcoat and is morally dubious, even though he's an angel.
Constantine movie also had a video game adaptation.
Riff of Sluggy Freelance fits most of the trope. His dark and troubled past working as a Hereti Corp freelancer is often brought up even today. His inventions potentially destroying all (or most) of mankind is another for which he's visibly trying to redeem himself for. Dimensional travel, hell's kittens, and time travel all rear their heads in this (supposedly) Slice of Life series. He also has a string of girlfriends he left behind. He doesn't smoke though.