Never mind if the powers could actually happen- Could someone with superhuman abilities actually use those abilities to fight crime…or would such a person be locked up and/or killed?
BTW, I’m guessing that knocking out bad guys, tying them up and leaving a note saying , “Courtesy of your Friendly Neighborhood Ladybug-Man” is totally out of the question.
Depends on the country. In most, I’d guess he’d be considered a vigilante and arrested, or at least the police would demand that he keep his deeds subtle enough that they could be passed off as the actions of the cops.
It would also depend heavily on what exactly his superpowers were and how powerful they were. If it’s invisibility or teleportation, for instance, he might never be caught.
Is it indecent exposure if they can’t actually see you? 
The person would have to take actions that do not depend on legal intervention. Much of their work would likely be negated in legal proceedings.
If the powers allowed. The person could disrupt illegal enterprises. Destroying contraband substances, goods etc. I am supposing the person is quite immune in some way from physical harm. Maybe they also have enhanced senses in order to spot such illegal things.
If a person invests some time and resources to the task. They may get access to some good information as to targets. They can be surprisingly obvious if you are not concerned with nailing them in the courts.
Hitting a few operations can remove the operators themselves, without one actually doing it yourself. Failure is often harshly dealt with in criminal groups. Suspicion among them can be incited to murderous outcomes. They take out their own trash.
Avoid public display in every way.
Spur of the moment encounters involving protection of civilians can be done. But within legal confines. Defending others or ones self is looked on kindly by authorities, if done in the confines of the law and without showing obvious powers.
Does one also have super self discipline and a pure moral code? Power corrupts.
Don’t wear a costume. A stocking mask is good. Mix up the wardrobe. Looking like criminal scum is best to mess with them.
This would be a human being from this world or, as was once said by Sebastian Toombs, “Even the Pope cheats at solitaire.”
Going further with the invisible angle, such a person (who decides to freelance entirely on his own) would instill intense fear in crooks, terrorists and bad guys everywhere. Putin would have no way to know whether he’s going to get offed in his sleep in his luxury bedroom. Every high-profile mysterious death would draw massive media coverage. It wouldn’t even be like Batman, for instance, who no matter what was at least visible to his foes. The difficulty would be him finding out where the bad guys are. Even the US intelligence community took many years to find bin Laden, for instance - unless we add superpowers of extreme knowledge to the equation as well.
I imagine working for govt like the Comedian or Doctor Manhattan from Watchmen would be more plausible than a vigilante lasting any significant amount of time.
~Max
In some lines of work, it is not always so good to enjoy it.
We tried that at Fort Meade from 1977 to 1995. It formed the basis of the non-fiction book The Men Who Stare at Goats (fictionalized in the 2009 movie starring George Clooney, Ewan McGregor, Jeff Bridges, and Kevin Spacey).
~Max
Assassination of criminals who have murdered, I feel is in bounds. Messing with governments? I would leave that to themselves in most cases. They put so much time and resources into it. It seems a shame to shortcut their efforts.
But this super person must know absolutely that the target has murdered.
That would still be against the law, even if the victim is a murderer him or herself, the vigilante would be arrested.
~Max
Not so sure about how long it took to find him. Maybe just a while to decide to end him.
Rabbit holes ahead…
As another such example, in Marvel comics, the Avengers have, for much of their existence, operated with some form of oversight/cooperation from the U.S. government.
It is against the law of course.
As is most of the things super heroes do.
OP asks if a real-life superhero would be arrested or killed, that’s why I made the point that assassinating murderers would lead to arrest.
@kenobi_65 I think Captain America was actually enlisted in the U.S. army for WWII. The first issue was him punching Hitler in the face, which I remember only because I read it pre-dates Pearl Harbor.
~Max
And governments always perform moral actions?
I think I would prefer to freelance.
I am assuming the super person could do it in such a way as to evade suspicion. Actual statistics on solving murders are not so great. And that is often with amateurs committing them. A super person should be able to evade detection.
If I had powers. I would use them discreetly with every intention of avoiding detection. Not fly in wearing my flamboyant costume and commit offenses for the adoration of the onlookers.