I've seen a few cakes with sugar stuck on the fondant, to give it that icicle look, particularly on winter themed cakes.
but I'm just wondering what it can be stuck on with and how long it would last? (As in, would it dissolve over time?)

I put sugar on a snowman cake - I just got the fondant moist where I wanted the sugar and "threw" the sugar at it! lol It adhered fine and stayed on no problem! I haven't tried thoroughly covering a whole cake though so I don't know if it would all stay on. You could use thinned corn syrup or piping gel though, those would probably do well.