First, definitions: For this purpose I mean a real superhero. Batman doesn't count, since he is merely a human with outrageous skill at fighting and stealth backed up by money and technology. Superman is a superhero, or the Flash, or Iron Man. These sorts of people are nearly invulnerable to a normal human and can do outrageous things that no person could aspire to. They are also hardly even human, or not human at all in the can of Superman.
So if you suddenly became one of these superheroes, what would you do? For example, a superhero can stop a mugging in an alley or fight terrorists without any moral quandaries. We would all pretty much agree these are moral necessities. However, a superhero could also fly into a country being run by a brutal dictator and assassinate him. A superhero could smash all the weapons of mass destruction in the world into scrap. A superhero could step into a longstanding conflict (think Serbs and Croats not so long ago) and massacre everyone involved in the fighting, forcing a bloody end to the conflict. Would these actions be moral?
These waters get very muddy. There is the issue that when an outside agency steps in to take control of a bad situation they often end up making things worse. Even if we could assassinate Kim Jong-Il (a clear example of a brutal, megalomaniacal dictator) it isn't remotely clear that doing so would make anything better. Surely someone else would step in to take over and there is no real reason to think that would improve the lot of those who live in North Korea. North Korea, among many other countries, could stand some fixing but the fixing it needs is democracy, education and freedom and those things are achieved by nations, not heroes.
The other really crucial consideration is that people don't deal well with intervention by a greater power. It is well documented that when a person wins a huge amount of money from a lottery they usually don't benefit much in the long run. The first little while is a mad rush to quit jobs, buy houses and go to extravagant restaurants and after a while the money is blown and the winner must go unhappily back to their old lifestyle. Contrast that to the effect of someone working very hard over time and accumulating a lot of money by dint of effort, skill and moxie. In those cases there tends to be substantial longterm satisfaction derived from that success. As they grow to maturity children learn to take responsibility for themselves and their future. People as a group are much like this; to grow up and begin making good decisions as a nation requires responsibility and an invincible watcher denies this.
In the end being a superhero might end up being more torture than joy. Your powers would make you capable of doing so much and yet by your very nature you would often cause more damage than you cure. I think very few people, thrust into the hotseat of being a superhero, could resist the temptation to use and abuse their powers. If there is a superhero out there somewhere hiding their powers from the world at large, I salute their wisdom and discipline. Humanity can become greater than it is but that growth requires responsibility, the existence of which depends on the lack of superheroes.