Doom & Destiny

Side quests are "given" automatically and simply require you to usually kill a certain number of enemy types, but may also require you to obtain items. You'll see your progress after a battle in which the specific enemy appeared.

Kill Quests:
Dead or Alive? - Kill skeletons
Ex-Orc-ism - Kill orcs
Angry Birds! - Kill birds
Hocus Pocus! - Kill mages
My name is Wolf - Kill wolves
Blue Screen - Kill cyborgs (I found them post game, northern map)
Double Drake - Kill drakes (I only found them post game)
Double Dragon -
Greedy! -
Mamma Mia! -

Collection Quests:
Wargiful! - Collect Warg pelts
Insert Cock Joke -
Winter Collection - Collect White Pelts (I found my first one post game, northern map via large boat)