There are few things more American than superheroes, and of course tabletop RPGs have plenty of offerings. However, as is often the case, only a few of them actually matter because only those few will readily get players to sit at the table. Here's that short list, and my takes on them.
Champions is the D&D of this genre in tabletop RPGs. Yet it is no class-and-level game. You have to build your man up from zero, literally using a points budget to design your hero as if he were a robot. The Game Master may level further fetters that constrict the mechanical operation of your man, and your man isn't that likely to change much over the course of play. There's a hell of a lot of dealing with rules up-front, and the combat system is slow as hell, but there's a reason it's still the dominant game (now in its sixth edition) in tabletop RPGs: it does deliver on its promise of doing any hero, allowing cross-genre play with ease. Its massive supplement catalog, and house setting, has a following akin to the One True Universe of Traveller, and many of the supplements are good (sub-)genre guides in their own right.
The number two game, and the only real contender for the top spot, is Mutants & Masterminds. This game arose as one of the breakout stars during the heyday of the d20 System, surviving the brand's collapse when Wizards of the Coast changed direction with D&D's 4th Edition. Currently in its 3rd edition, it's been used for the recent version of DC's licensed game. It too offers to handle any hero, and it has not failed to deliver yet, for much the same reason as Champions: it's a point-build system, where "level" is just a power limiter. While simpler (by comparison), that also means fine mechanical detail is lost (but nothing a competent Game Master can't handle).
The only class-and-level game on this list is Palladium's Heroes Unlimited, and it's always going to be no better than #3 because of two big flaws right out of the gate: it can't do any hero (Hell, it can't even do Iron Man!), and its rules are poorly presently to the point of actual contradiction (as in "There are contradictory character generation rules."), requiring that the Game Master either find errata (Good luck!) or fix it before he sits players down. It's still a better option than many you'll find, with only true classics like Villains & Vigilantes giving any competition.
What this comes down to is that the medium is perfect for "Who would win?" questions, then to answer "But this is how it should be done!" fanboyisms, but the lack of clear and sufficiently-frequent character progression (ala D&D) torpedoed the appeal alongside far too much player-facing interaction with rules and mechanics up front. Until these, and other known issues, get fixed you'll continue to see superhero RPGs languish worse than D&D and its clones (and similarly not succeed in videogames for similar reasons).