Who you gonna call?

I don't know of any fictional detective that doesn't solve murders so it probably doesn't matter which one I choose. Therefore I'll just pick my favorite: Mr. Holmes. But, I mean the real Holmes -- the one as written by AC Doyle, not any of the Hollywoodified versions that came later.

My second choice would be C. Auguste Dupin, the detective in Poe's "The Murders in the Rue Morgue" and other stories. Poe basically invented the detective story and Dupin was his sleuth. Dupin was created even before the word detective was coined.