Any and all.
You can also get seals by doing the quests at your GC's front desk. Repeatable, but on very a long timer.
FATEs are simply the fastest way to accumulate seals (and non-quest exp) at the moment. Several hundred seals for the 20-30 range FATEs.
What are the GC quests from the desk like?
And do you know if they still have the gathering/crafting item dumping for GCs now?