Q:

Fate of Norns and their web

I am wondering about one thing: What is the fate of Norns and their web during the Ragnarök?

A:

Unknown. Our main sources for Ragnarök are the Völuspá, the Vafþrúðnismál, and what Snorri retells of these. The Norns are introduced in verses 19-20 of Völuspá, but that is before Ragnarök starts, and they are not mentioned again. They don't seem to be mentioned at all in Vafþrúðnismál. Snorri does not provide more information.
At a guess? Since we know that some gods and goddesses, as well as some forces of evil, are going to survive Ragnarök, it would be reasonable to suppose that the Norns will as well.

