I'm not sure if the names of these tracks are correct, but I think most of them are.

Title
Used in:
47th Street
Level 1 Part 3, Level 7 Part 1
Always was a Big Mouth
Pre-first boss fight, pre-fifth boss fight
Attract
Used in the first attract mode sequence
Attract #2
Used at the end of the first attract mode sequence
Attract #3
Attract mode character bios, Level 6 opening cutscene
Behold True Genius
Level 4 opening cutscene
Boss Battle #1
Fight against Vice, Morgan/Morque and Fessenden
Boss Battle #2
Fight against Butcher and Tyrog
Boss Battle #3
Fight against Hogg, Pre-credits sequence
Boss Battle #4
Fight against Slice and Slisaur
Cave
Level 8 Part 3
Continue
First 10 seconds of the continue screen
Continue #2
Last 10 seconds of the continue screen
Continue #3
Used when inputting your initials at the continue screen
Eastcoast 2513
Level 1 opening cutscene
Elevator Down
Level 7 Part 2
Fessenden Introduction
Pre-final boss fight cutscene
Fessenden Defeated
Pre-level 7 boss fight, Fessenden's death
Game Over
Game over screen
Going to Jack's Garage
Level 3 opening cutscene, Level 7 opening cutscene
Hell Road
Level 3
Hello this is Jack
Level 5 opening cutscene
In the Building
Level 1 Part 2, Level 6 Part 1
In the Garage
Level 4 Part 2
In the Garage #2
Level 4 Part 3, Level 7 Part 3
Jack's Garage
Level 4 Part 1, Level 8 Part 2
Mud Swamp
Level 2 Part 2, Level 6 Part 3, Level 7 Part 3
Player Select
Character select screen
Power of the Doctor
Pre-second boss fight, Pre-sixth boss fight
Round Cleared
Whenever you beat a level.
Staff Roll
Credits sequence
Swamp Forest
Level 2 Part 1, Level 6 Part 2
The Underworld
Level 8 Part 1
This Doctor is Mad!
Final boss fight, when Fessenden has transformed into his ultimate form
Top of Building
Level 1 Part 1, post credits sequence
Unused Track
This track is not used in the game, to my knowledge.
Village of Flame
Level 5 Part 1