After the destruction of Sauron's One Ring, what happened to the other 19 rings of power (3 Elven, 7 Dwarven, 9 Men)? Did they become powerless, remain cursed, become cleansed, disintegrate, or something else entirely?
The Three are still around, and it is debatable whether they have any magic left.
Sauron had long ago obtained control over the Nine, and used them to control the Ringwraiths. It is never made clear whether they were in Sauron's possession or worn by the Wraiths (in The Lord of the Rings, Gandalf states both that “the Nine he has gathered to himself” and “the Nine the Nazgûl keep”; in a 1963 letter to a reader, Tolkien mentions “their nine rings (which [Sauron] held)”, which may have refered to either mental control or physical possession).
Of the Seven, four were destroyed by dragons, and Sauron obtain the remaining three. (Source: Gandalf's exposition to Frodo when he reveals that Biblo's bauble is in fact the One Ring.) The last of the Seven was the one held by Thráin, the father of Thorin, who died in Dol Guldur (then the home of the Necromancer, who was later (post-Hobbit) revealed to be Sauron) shortly after meeting Gandalf, setting in motion the events of The Hobbit.
The Lord of the Rings makes no mention of the fate of Seven or the Nine after the destruction of the One Ring and of Sauron, not even in the appendices. In the last chapter of the Silmarillion, it is only mentioned that:
the One and the Seven and the Nine are destroyed; and the Three have passed away, and with them the Third Age is ended
It is likely that the Nine and the remaining among the Seven were destroyed (or at the very least that their magic was lost) when the One that controlled them was destroyed, or when Sauron was consequently destroyed.
Well, the three Rings went over the sea with their bearers (see last chapter and memory), and the Dwarven Rings were partly already destroyed by Dragons.
That leaves us the Nine and some Dwarven Rings.
My personal view is that they were destroyed along with everything else made (partly) by Sauron; after all, even the Towers of Teeth (originally made by Gondor) fell down, and Sauron played a big part making the Rings.
What happened to the Nazgûl was this:
And into the heart of the storm, with a cry that pierced all other sounds, tearing the clouds asunder, the Nazgûl came, shooting like flaming bolts, as caught in the fiery ruin of the hill and the sky they crackled, and went out.
The Nine Rings probably perished with the Nine Riders, or maybe the Nazgûl were destroyed because of the Rings...