Welcome to the Downs, Gorhilion. Enjoy death!
To answer your question, if I remember correctly, the One Ring could not be destroyed by anything save the Fires of Orodruin. The other 19 Rings, 3 for elves, 7 for dwarves, and 9 for men, could be destroyed by dragonfire. I think only four of the dwarf Rings were, but none of the others. I also don't think there were many dragons around during the Third Age.