[Boromir88]

As far as the Three Elven Rings, yes they hoped that they would still retain their powers after the destruction of the Ring, but they certainly feared that if Sauron found the Ring, or if the Ring was destroyed the powers of the Three would lose their powers:

Yet many voices were heard among the Elves foreboding that, if Sauron should come again, then either he would find the Ruling Ring that was lost, or at the best his enemies would discover it and destroy itl but in either chance the powers of the Three must then fail and all things maintained by them must fade, and so the Elves should pass into the twilight and the Dominion of Men begin.~Of The Rings of Power and the Third Age

And eventhough Sauron never "sullied" them they were still controlled by the One Ring:

Therefor the three remained unsullied, for they were forged by Celebrimbor alone, and the hand of Sauron had never touched them; yet they also were subject to the One.~ibid

Before I get to the Nazgul, and I think with the Nazgul it's more up for grabs as to what happens to them. I just wanted to point out, excellent post Farael, with some good ideas, just one thing that I have to point out because it annoys me.

Let's not forget that they were all kings in life.

This is actually a fault caused by the movies, they all weren't "Kings of Men." After they received their Rings:

Men proved easier to ensnare. Those who used the Nine Rings became mighty in their day, kings, sorcerers, and warriors of old.~ibid

So after they got the nine they became kings, sorcerers, and warriors. Before, the most we are told were that 3 were Lords of Numenor:

It is said that among those whom he ensnared with the nine rings three were great lords of Numenorean race.~The Akallabeth

So what would happen to them if the One Ring was destroyed? Well it appears to me that Sauron (who at some point in time took back the 9 rings and was in possession of them) could still hold control over the Nazgul, eventhough he did not hold the One Ring:

They would have obeyed or feigned to obey any minor commands of his that did not interfere with their errand… That errand was to remove Frodo from the Crack . . . They would have greeted Frodo as ‘Lord’ . . . and induced him to leave Sammath Naur to look upon his new kingdom…~Letter 246

So Sauron could still control the Nazgul eventhough he did not have the One.

When the One was destroyed I had assumed the Ringwraiths were also destroyed. What happens to their spirits afterwards I don't know if it's ever said (perhaps someone else does?) But I've always assumed they were destroyed when the One was.

And into the heart of the storm, with a cry that pierced all other sounds, tearing the clouds asunder, the Nazgul came, shooting like flaming bolts, as caught in the fiery ruin of hill and sky they crackled, withered, and went out.~Mount Doom