Let's make it simple. The coolest superhero has to be one of the Big Four: Superman, Batman, Spider-Man, or Wolverine. Nobody else comes close in terms of popularity, and if you're not the most popular, you can't lay claim to being the coolest (this may seem unfair, but superheroes are geared to young people, and young people decide what is cool by virtue of what is popular). This way, we eliminate also-rans like Iron Man, Hulk, The Green Lantern, Captain America, Thor, Ghost Rider, The Flash, and The Punisher (as cool as any of them may be on their own merits, they're just not iconic enough).
II: Define this
I don't know of any better way to define coolness than "The ability to make me want to re-make myself in your image". There are a lot of factors that go into this feeling of inspired envy, but that's the definition in a nutshell.
SHORT VERSION -- If you're cool, I SHOULD want to be you. If I don't, then what's the point of being cool?
III: How they stack up
SUPERMAN -- Most powerful being in the universe. Flying is cool, going by my established criteria. Some of his powers seem redundant or unnecessary (X-ray vision? WHY?). His origin story is cool (everybody wants to be the last of an extinct race). He chooses to pretend to be a huge dork and weakling as his alter-ego, for reasons that are mysterious to me. He has a day job as a newspaper reporter, which is certainly not cool. Hangs out with his co-workers, which could be cooler, but isn't too pathetic by normal standards. His main enemy is a genius who's bald, so whatever. Oh, and his weakness is a green rock. LAME.
BATMAN -- Rich kid with fascist streak. His weapons and vehicles are cool. He's an orphan, and who hasn't wished that their parents were dead? Unfortunately, he mainly only hangs out with a teenager and an old butler, and loses major cool points as a result. His arch-enemy is a psychotic clown, and anybody who beats up clowns is cool. For his alter-ego he pretends to be a hedonistic playboy, which would be cool if it wasn't so divorced from reality. Doesn't really have a weakness, unless you consider "dressing up like a bat and fighting bad guys" to be a weak move in itself.
SPIDER-MAN -- Does whatever a spider can. Often cracks wise while cracking jaws. Started life as a nerd and crawled his way up to public enemy. His personal life has too many problems, and unfortunately those parts of the story often overtake the superhero/fighting evil stuff. His arch-enemy is Green Goblin, which is weird. Hangs out with his super-old aunt and a redheaded chick. Works at a newspaper for an a-hole of a boss. Almost nothing about Spider-Man is cool (he is too much like you and me), but that's what made him a groundbreaking character at the time.
WOLVERINE -- All-around bad-ass. Rips fools to shreds and then casually walks away. Can't be killed. Lives longer than everyone. His arch-enemy is Sabretooth, who was originally written as his father (and the only thing cooler than wanting your parents dead is actually attempting to murder them yourself). Actually, maybe his real arch-enemy is Cyclops (who's a huge douche, so either way he wins). He hangs out with a bald dude in a wheelchair and a bunch of confused, angsty teenagers who he acts as mentor to. His day job is sitting around in bars and getting into fights with patrons who are stupid enough to antagonize him. Has awesome hair. Nothing about this guy is not cool.
Wolverine is the coolest superhero ever. His powers are cool, he has cool friends, and his approach to life is pretty much the epitome of coolness. He clears absolutely every important category of coolness, and although some of the other guys give him a run for his money in certain categories, they also come up lacking in others. You could come up with good reasons why little prepubescent boys would not want to be Superman, Batman, or Spider-Man, but there's no reason why any of them would not want to be Wolverine in a second.
That was easier than I thought. Next time I will be examining the question "Who is the gayest superhero?"