Someone posted a list of all the available background music on GameFAQs. You can set music based on either stages or make it character-specific:
Turn On the Lights
Russian Roulette
Machinations
The Roar of the crowd
Skyscraper
Beginning To The End
Scream
Polluted Air
Fire Alarm
Omen
Liberation
Collapse
Rapids
Cascade
Primitive Rhythm
Oblivion
Like a Dragon
The Secret of Kung-Fu
Tedious Days
Accelerator
Happy Funny Crazy
Tiger's Family
Light the Fuse
Niug-Nep Place
Into the Abyss
Tradition
Graceful Sounds
Gotta Move on
Knock You Down
Superstar
Mist Knows
Demensions Training
Can't Stop This Feeling
Urges
Repetition
Wired
An Invisible Shadow
Showdown
Cheerful Buddhist Priest
STAGE
Zany Zapper
I Feel For You
Fang
Hunter's Moon
Purity
Mentor
Solitary Desperation
Steel Horse Rider
Moonlit Lady
The Wind is Howling
Flowerbed
Shade
Successor
Tribal Beats
Alpha-152
Drunken Wolf
Serpent Dance
Turn On the Lights
Russian Roulette
Machinations
The Roar of the crowd
Skyscraper
Beginning To The End
Scream
Polluted Air
Fire Alarm
Omen
Liberation
Collapse
Rapids
Cascade
Primitive Rhythm
Oblivion
Like a Dragon
The Secret of Kung-Fu
Tedious Days
Accelerator
Happy Funny Crazy
Tiger's Family
Light the Fuse
Niug-Nep Place
Into the Abyss
Tradition
Graceful Sounds
Gotta Move on
Knock You Down
Superstar
Mist Knows
Demensions Training
Can't Stop This Feeling
Urges
Repetition
Wired
An Invisible Shadow
Showdown
Cheerful Buddhist Priest
STAGE
Zany Zapper
I Feel For You
Fang
Hunter's Moon
Purity
Mentor
Solitary Desperation
Steel Horse Rider
Moonlit Lady
The Wind is Howling
Flowerbed
Shade
Successor
Tribal Beats
Alpha-152
Drunken Wolf
Serpent Dance