Really depends on what the powers were.
A lot of powers could prolly be used more constructively than crime fighting, and lots of offensive combat oriented powers can't really do shit weapons can't accomplish already.
Whats the big deal about beeing able to shoot fireballs in a fight if you could just use a gun?
Lets take spiderman for example, while the spider sense and über physical skills are nice, he would fare way betetr in combat if he used a gun, and he'd prolly be dead several times if his enemies could actually hit shit with their weapons.
And in case of vigilantee justice, if the police wanted to captrue him, they would prolly find him out eventually.
Generally, if your powers doesn't offer you invulnerbillity/resistance or some otehr emans to protect you from gunfire, you're not that much at an advantage really, and you're prolly not gonna be a hero for long before you'll become a dead hero.
So spuerman-ish level powers, sure fight crime, don't give a shit about what people think, not like they can do shit anyways, but smaller advantages like most superheroes, i'd prolly find a safer way to use them, and let the police handle what they're already betetr at than i would with my newfound powers.
If it was powers that could be used to fight crimne in a less direct combat way (think professor X, invisibillity etc) i'd prolly ask tyhe police if they could use my powers, and if not find someone else who could.