I guess I'm horribly picky about my heroes; minor imperfections aren't enough, nor is some sort of overplayed tragic past. If they want the "Hero" tag, they are going to earn it
in the strongest, perhaps most morally-questionable fashion possible.
I want my heroes to face the darkest sides of themselves--to see within them their darkest capabilities, to confront the monster within. To see with that monster's eyes the darkness inherent in their particular universe. And from there, I want them to decide what they will do with it. Will they rise completely above that darkness--while continuing to acknowledge it and accept it as part and parcel of what they are? Will they try to twist some of those elements and use them for the greater good, even if it means indulging a "scarier" part of themselves? Or does the monster win the conflict, taking an unassailable hold over them?
I've seen heroes of all three categories, though the last would probably be termed the "antihero", in a more accurate sense. Naturally, your mileage may vary.
It's tough to write a story with a good, strong hero. Good luck!