Brenden Fletcher's run on Black Canary may be of interest to you. It reads like a cross between Scooby Doo and Scott Pilgrim. Dinah is less a superhero and more a woman trying to hold down a music career, yet inexplicably getting mixed up into martial arts hijinx that involves cults, monsters and other creeps.
You may also enjoy Constantine the Hellblazer. The comic leans more toward horror, but John has plenty of adventures solving mysteries, hunting for artifacts and sending demons back to hell.
If all else fails, look into the Vertigo imprint. All of the more interesting, non-superhero stories are published under that imprint. I recommend Y: The Last Man.