Militant music--songs sung during battle, about battle, after battle. High quality military music might give dwarfs a morale boost, for the Army Arc.
Ofcourse, singing songs can also give away your unit's location...so it wouldn't be useful in every situation.
Songs of grief, aka durges, seem appropriately "dwarfy".
Berserk dwarfs might also begin to sing. Possibly a "death-song"?
Dwarfs might compose songs for other dwarfs, either as tribute, as part of a mating ritual (serenading each other), or songs of grief, about a loved one or a hero's passing.
Rich dwarfs/Nobles might hire other dwarfs to perform.
"Street musician" type dwarfs (and other species) might earn money by donation from passers-by.
Work songs have an extremely long history with humans, and I think dwarfs should have them.
Dwarfs might make music with tools (hammers and such), as well as specific "musical instruments".
Artifacts could occasionally sing
(the Singing Sword?)
Dwarfs might have certain songs they sing during specific holidays (Christmas music, anyone? Auld Lang Sine?)
Sometimes, these could be military music (4th of July), while other holidays could be more peaceful.
Singing/Music might also be used during various rituals, magical or otherwise. Artifact songs might be magical.
There might be times and places where singing is inappropriate (at night, while other dwarfs are sleeping).
Dwarfs might have specific songs that are their favorites. Others they dislike.
Songs could be traded between species, and altered, to fit a different species' sensibilities and values. Songs could just change, over time.
Dwarfs could be very aware of acoustics, due to their architectural knowledge, and living underground. Songs might be of "higher quality" in a place with great acoustics.
Sound could be a weapon (Foa's banshees, also sonic breath?).