I don't really listen to music unless there's a weather change in-forest, like rain... I should probably try it more often. XD; But to me there's something wonderful about the atmosphere change and the rain in the background of the song(s).

For rain, I love listening to Sigur Rós (example) or any of their side-projects. I also love Clint Mansell (example).
The next time there is fog, I'm going to try Burial (example), I stumbled across them about a month ago and have been waiting for a weather change to try them out. :3

I tend to avoid listening to music since it can create a bit of lag in-Forest.
But the sounds in Spyrre's bio can really add a lot to the Forest's sounds. If I click on it, I tend to leave it open for awhile just to listen.

I tend to listen to movie soundtracks when I'm in-game. :')

Edit; Or, of course my favourite bands; Nightwish, Gothminister, Van Canto or Kamelot. :b

I tend to listen to music that goes with the character I'm currently playing... if that makes any sense. But it's always something calming, with no lyrics.

Most of the time I just enjoy the chirping of the birds and such, however.

Haha I tend to listen to whatever I want when playing the forest. The music usually effects me/my character's mood and the way I play.

If you want something calm instrumental piano-y you could check these out:

Isaac Shepard
Akira Kosemura
Ólafur Arnalds
Michael Andrews
Whitetree
Silver Mercury - A Lonely Saturday Night
Aaandd obviously The TEF Tunes!

I just listen to whatever I usually listen to xD everything, in other words. usually with spotify. I haven't even thought about adjusting my music to the game.

I listen to instrumental music a lot. Particularly movie soundtracks. These are a few of my favorites, you should give them a listen if you get the chance.

-Arrival to Earth by Steve Jablonsky
-Mulan's Decision by Jerry Goldsmith
-Driving with the Top Down by Ramin Djawadi

I also listen to songs in other languages. The lyrics don't distract from the forest ambiance.

-Praan by Palbasha Siddique (Composed by Garry Schyman)

My (though not technically mine) music depends of how I feel, what character I'm on, stuff like that. Now watch, as I name bands and songs that I like, and shall mock my own taste in music by commenting on them.

The Polyphonic Spree <----- You don't get more original than them XD
Matt & Kim <----- You can't get MORE intrusive music anywhere else
Aqualung <----- Because warbling is the key to success.
Prayer by Hayley Westenra <----- I won't complain about this song. It's perfection in a cup.
Man's Road by America <----- From the Last Unicorn. Of COURSE.
Apricot Stone by Eva Rivas <----- It's literally a song about an Apricot Stone.
Sweet People by Alyosha
Modest Mouse <----- Listening to drunken men singing is the best kind of singing
Lenka <----- Okay, I can only say that her some of songs sound normal. And that's what I hate about it.
Oren Lavie <----- Even total scruffbags can sing
Corner by Allie Moss <----- I swear that song is about Epon :S

I would bore everyone further but I think a list of 11 is long enough. My music is, personally, awesome. Fact.