Random+Super+Powers

So I've had an idea bouncing around in my head for a while now. But I don't know if anyone else would be interested, or if it's just me. Or what the best way to handle it would be.

Basically my idea is that the characters would have super powers, but they wouldn't be stable. Instead of just one super power each character would have a pool and when they triggered them, either consciously or because they were under stress, one of those super powers would be randomly selected from the pool. Which means they might not get the power most applicable to the situation. For instance getting the ability to manipulate fire when you're trying to escape a collapsing building and really wanted super speed. Then the problem would be how to use the power they did get most effectively.

A few variations I have thought of.
 * Characters can try and influence which power they get. It would still be random, but more likely to trigger the one they want. At least as long as they were successful in whatever mental check the choosing required.
 * All the characters share one pool. If one character activates a power then that power is not available to another character until the first one deactivates it.
 * The individual powers grow stronger with use, but suffer from increasing cooldown periods or other penalties.
 * Some of the powers are downright negative, or at least uncontrollable. For instance the characters body starts spitting out energy bolts in random directions. Potentially that could be useful in a fight where you're heavily outnumbered. Everywhere else it would present a hazard.

The actual setting for a game with this I haven't settled on yet. I have a few ideas, but mostly I want to see if the core mechanic I am suggesting would be interesting. Or even playable since there may be issues I am overlooking. Just to put them out there here are a few of the settings I am considering.
 * A city in which supers are if not common at least not unknown. The city may range from medium sized all the way up to continent spanning hive city depending on how things go exactly. Enemies would range from super villains to mega-corporations to organized crime to governments to the citizen's association which thinks supers are a menace. Players could more or less choose what they wanted to do. Or I could make some overarching plot.
 * An after the end scenario where the apocalypse has triggered mutations and energy absorption which caused the powers and their randomness. This setting would involve facing hordes or ravenous mutated beasts and other threats. Players would have the option of trying to rebuild civilization if they wished, but that wouldn't be the only choice.
 * A far future game where the characters are explorers looking for new useful systems either for an employer or independently. This would involve exploring new worlds and dealing with the dangers they might hold. It might also involve dealing with space pirates and employers who don't want to pay you for what you find.

Right now I think GURPS would be the best system to use for this as it has plenty of power options and it would be relatively easy to build powers that could be switched out with each other. But I'm open to other suggestions if something else would be more suitable.