Yes, I often try, however, because I do fall asleep to my music, I don't stay asleep... That is, when the songs change, I wake up.
If there was a loop of whatever I was listening to, then I could probably stay asleep. The only time is can change without me waking up, is when I've reached that deep-sleep stage.
The best music for me to fall asleep to is probably something like Marilyn Manson, or KoRn. (simply because I love the music)
And yes, I can even fall asleep to Five Finger Death Punch. Metal is melodic to me... and especially certain tracks of MM or KoRn. They tend to have at least one or so tracks that are more fluid, and slower than the rest. Perfect example, the songs "Tearjerker", "Kiss", and "Hallow Life" by KoRn, --or "The Speed of Pain" and "♠" by Marilyn Manson. Those are more melodic in comparison to the rest of the artists' songs. The song "Alone I Break" also seems to have a powerful effect on me, and also can lul me to sleep.