Feathers can be obtained in two ways:
(1) Breaking Claws or Saddles
(2) As the reward for completing a bonus dungeon

In the case of the former, there is a strict set of requirements.
(1) The Claw or Saddle must have been combined (has blue font.)
(2) The Claw or Saddle must be at least plus one (+1.)
(3) In general, each claw or saddle yields two different feathers, one a lesser +X and another for a greater +X.

As for getting feathers as a reward, here's the rundown on that:
Chocobo's Dungeon: Random Feather of any kind
Mog's Dungeon: Random Summon Feather
Shiroma's Dungeon: Random Red and Yellow Spell Enhancing Feather
Kuz's Dungeon:
Cid's Dungeon:

All feathers can be obtained except for one by breaking claws or saddles.  That feather is the Energy UP Feather.  However, all feathers can be obtained through the bonus dungeons.

The only other feather-pertinent area is the Tree of Life in the village.  As far as I know, the only prize you get for sacrificing your feathers for the tree is a random "good" nut.  To me, this is by far, completely worthless.  If anyone has any solid evidence that there really is a "good" point to the Tree of Life, email me.

Red and Yellow Feathers / Spell Enhancing Feathers

Blue and Purple Feathers / Status Boosting Feathers

Reddish Feathers / Summon Feathers