The follwoing is a list of PS3 PSN games (the downloadable stuff) in 
which should allow you to listen to your own music stored on a 
PlayStation 3 while also playing the game...

Most of this information comes from 
https://www.playstationtrophies.org/forum/topic/267719-games-with-custom-soundtrack/
with a tiny bit of editing for easier reading.

------------------------------------------------------------------------

Custom soundtrack can be played in two ways:

XMB: this is done by pressing the PS button in the controller during the
     game, then scrolling to "Music" and select any track/playlist that 
     is saved. It should be noted that some games (mainly, during online
     multiplayer) will automatically pause the music, any attempt to 
     resume it will pop a message saying: "Music cannot be played at 
     this time, it will start automatically in a while", these games 
     will be labeled with "offline only".

Menu: some games can play custom soundtracks by going into a menu 
      (usually, it's under "audio options", but some games can work 
      differently) or by creating a playlist (some games specify how to 
      name a playlist in order to be recognized). I will go into further
      details in explaining how to set up correctly the custom 
      soundtracks.

 XMB

After Burner Climax
Anarchy Rush Hour (Offline only)
Armageddon Riders
Astro Tripper
Back to the Future: Episode 1
Back to the Future: Episode 2
Back to the Future: Episode 3
Back to the Future: Episode 4
Back to the Future: Episode 5
Bejeweled 2
BlazeRush (Offline only)
Burn Zombie Burn
Call of Duty: Classic (Offline only)
Comet Clash
Crash Commando (Offline only)
Crazy Taxi
Critter Crunch (Patch is required and single player only)
Crystal Defenders
Cubixx HD
Cuboid
Dead Nation
Deathspank: Thongs of Virtue
Digger HD
Diner Dash
Dust 514
Dynasty Warriors 8: Empires - Free Alliances Version
Echochrome
Fancy Pants Adventures, The
Feeding Frenzy 2
Folding@Home (software)
Flock
Funky Lab Rats
GundeadliGne
Gundemonium Recollection
Hamsterball
Hitogata Happa
Hustle Kings
Hydrophobia: Prophecy
Hyperballoid HD
Interpol: The Trail Of DR. Chaos
Invincible Tiger
Legasista
Magic Ball
Mahjong Tales: Ancient Wisdom
Mainichi Issho
Marvel Pinball
Marvel Vs. Capcom 2
Metal Gear Solid: Peace Walker HD
Might and Magic: Clash of Heroes
Military Madness
Minecraft: PlayStation®3 Edition (Offline only)
Motorstorm RC
Noby Noby Boy
Nova Strike
Numblast
Peggle (Offline Only)
Penny Arcade Adventures Episode 1
Penny Arcade Adventures Episode 2
PixelJunk Eden (Upon completion of the game)
PixelJunk Monsters
PixelJunk Shooter
Pixeljunk Sidescroller
Planet Minigolf
Plants vs Zombies
Revenge of The Wounded Dragons
Sam & Max: Beyond Time and Space - Episode 1
Sam & Max: Beyond Time and Space - Episode 2
Sam & Max: Beyond Time and Space - Episode 3
Sam & Max: Beyond Time and Space - Episode 4
Sam & Max: Beyond Time and Space - Episode 5
Sam & Max: The Devil's Playhouse - Episode 1
Sam & Max: The Devil's Playhouse - Episode 2
Sam & Max: The Devil's Playhouse - Episode 3
Sam & Max: The Devil's Playhouse - Episode 4
Sam & Max: The Devil's Playhouse - Episode 5
Scourge: Outbreak
Section 8
Simpsons Arcade Game, The
Skullgirls
Smash Cars
Sonic the Hedgehog 4: Episode 1
Soulcalibur: Lost Swords (Servers are closed, it works after pressing 
                          start in the title screen and leaving it there
                          with the message "Service has ended")
Swarm
Tank Battles
Tetris
Tony Hawk's Pro Skater HD (Offline only)
Under Siege
Voltron: Defender of the Universe
Wakeboarding HD
Weekly Toro Station
Zen Pinball (Offline only)
 

Menu

.detuned (Press Start > Change Music)
Alien Zombie Mega Death
Beat Hazard Ultra (During Stages only)
Dress
Eat Them!
High Velocity Bowling
Mercury HG
PAIN
Savage Moon
Supersonic Acrobatic Rocket-Powered Battle-Cars
Super Stardust HD