As spoiled by the picture I chose for the post, my pick is Katana Zero.
Each stage has its own track, and it is not just background noise. Zero carries with him a walkman, and each level begins with him pulling it out at putting his earphones in, then starting the track as the name of the song appears on screen.
Whenever Zero isn't himself listening to something, any music heard is environmental, like the soft background music of a hotel lobby, or the annoying low tones passing through the walls from a party at the neighbors.
If you're a fan, and didn't know, there is an expansion coming to Katana Zero. It's been teased with a few gameplay clips, and a new OST track.