I can't really remember all of them, though I will try my best.
(not in order)
- there's a quest in Guardia Town that will reward you with one of the elemental monkeys.
- there's one in Highpoint Town that is related to the Ruby Destiny Fragment.
- there's one in Rintah Town about the symbol/badge of the town where Champion Joseph gives you.
- Blitz will give you a quest that is related to the Ruby Destiny Fragment.
- there's one in Silver Desert south from Silver Town and north from Rainleaf Town. It's about a guy who wants to see the treasure inside the Mirage Tower.
so far, those are the only ones that I could remember...