I get a lot of letters so I don't keep most of them, but I do keep the funny ones as they make me smile every time I read them. It's amazing what they come up with sometimes.
Oh, and I try to keep for awhile the ones where they say that they had a horrible dream and you left, and then they want you to promise you'll stay around forever... *sniffles*
I have used my primary character to keep the "moving out" letters from two villagers, Velma and Walt. I've had the game since August, and those are the only two I did not keep.
I keep some of the funny ones and then the letter they send when they move out of town. I attach furniture, etc. to the letters and use my mailbox as storage.
Like most here, I keep moving out letters. Haven't gotten a birthday one yet, but I think I might keep those, too. From my dreamies, I also keep the first letter since I send them a welcome one and they actually respond to that. But I only have one dreamie so far, so not a lot of letters, haha.
Back in the Gamecube version, I kept every single goodbye letter. It let me remember who I've had and it makes me feel nice inside when I read them, though it did bring up occasional sadness at them leaving.
But since then, I have thrown out every random letter or mom letters, gosh i hate those. But I keep every single goodbye letter!
I keep some of the Nintendo letters, a lot of the mom letters, Gulliver letters, and the Katie letters. But I make sure that they're all unique so I won't clog up the space.
As for villager letters, I had to cut back on them. So I keep the really sweet or funny ones, letters from a villager's birthday, and moving out letters. However, I only save five letters from each villager I've ever had. It's really saved me a lot of space.