Well, dunno what superhero couldn’t wrestle a polar bear…but I know for a fact that the Black Panther could. That’s 'cuz in “the Mighty Avengers” #169, the Black Panther wrestles…a polar bear! (Can’t find a larger image of the cover, but believe me, there’s a polar bear lunging at T’Challa on it).
Obviously a mutant polar bear with human intelligence, intent on defeating T’Challa and gaining control of Wakanda’s priceless vibranium!
Depends on your definition of “super,” I guess, but these guys would have some problems, I think:
The Riddler
The Joker
The Penguin
The Kingpin (maybe – I’m not sure how well judo works against bears)
Two-Face
Catwoman
Shadowcat
Nightcrawler
Angel (maybe he could find something really big to drop on it)
John Constantine
Scarecrow (unless his gas works on animals)
Robin
Iron Fist
Lex Luthor
Phases bear into ground and leaves it there; instant death.
Hops onto bear and grabs head, then teleports away with it; instant death.
:eek: Summons Iron Fist and SPLAT ! ! Bear bits everywhere !
Doesn’t he normally have a gun of some sort ?
I’m surprised nobody’s mentioned Frog-Man or the Spectacular Spider-Kid.
I would like to point out that only one hero I’m aware has actually done what the OP suggests. Doc Savage killed a polar bear with one punch then tore steaks out of his hide with his bare hands! No super-powers.
Batman’s a pussy next to Doc.
What about Bishop? Ooo, he can absorb energy…not very useful when there are huge claws in your guts.
Okay, we did specify earlier that normal equipment is allowed, right?
Stalemate. The bear can’t solve riddles, and would never find him in the first place. 
Oh, you know he has a contingency plan against all cold-weather animals. Yes, I’m aware that penguins are antarctic, still, it fits. Besides, some of his umbrellas are bretty nasty.
Same as Batman really. Belt full of knives, drugged up darts, explosives, pepper spray, etc. Enough to send a bear packing, if not kill it.
In a buisness suit? No. In a Kryptonite powered battle suit? Yes.
While (almost) any superhero can tote around high-powered firearms, most don’t. And the OP restricted us to a superhero’s normal gear. So we’re allowed to say the Punisher gives U. horibillis an acute case of lead poisoning, but we’re not allowed to say that Batman does the same thing (though the Bat probably does have other equipment which would do the job).
On the “you’re not a real super unless…” criterion, I think we ought to also make an exemption for specificity. Someone like Mimic (or was he called “The Bus that Couldn’t Slow Down”?) whose powers involve interaction with the powers of others can be quite formidable, in the right circumstances, but a fight with a wild animal just happens to not be one of those circumstances. He could, however, beat most of the folks who can beat a polar bear (or at least, give them a good run for their money).
Actually, not to continue busting your hump over the whole Mimic thing, the original Mimic ended up permanently mimicking the powers of the five original X-men. So in that incarnation he could easily take down a polar bear. Telekinetically hold the bear in place a la Marvel Girl and then use Cyclops’s optic blast to take its head off.
I don’t know how current the Mimic (Exiles) information is but if the linked entry is still correct he’d still be more than a match for the bear. He’s got half of Northstar’s speed, is half as strong as Colossus, has half-power optic blasts and a half-strength healing factor.