I think it would depend heavily on how many people got superpowers, and the scale of those superpowers. If everyone got them, but they were just convenience level superpowers, there would probably be a tonne of research done into them, but things would just go back to practically how they were before.
If there were few people who got superpowers, but they were very strong, there would probably be a lot of fear and distrust of them from the general populace. Some of them would probably overthrow their governments (or try to), and jobs like bouncers, security and policeman would suddenly become incredibly dangerous (much more than they are already, that is) because any criminal or person causing a disruption could possibly have the capability to level the building. It might even become a shoot first, ask questions later policy. Again, lots of research done, but there would likely be few willing subjects, and experimenting on/researching into unwilling subjects would be incredibly dangerous.
If everyone got a superpower and the power of the powers varies wildly, there would probably be major chaos. I think most governments would collapse unless they were lucky enough to get some truly high-powered individuals on their side. I think it would take at least a decade, maybe even as much as a century before the world got back to an ordered state, if ever.