Here's as much music from the Final Fight series and related games as I can get. Everything is in MP3 format.

= MIKE HAGGAR AWESOME. I really like this track.

[Capcom Vs. SNK] [Final Fight] [Final Fight Amiga] [Final Fight CD] [Final Fight Double Impact] [Final Fight One] [Final Fight SNES] [Final Fight 2] [Final Fight 3] [Final Fight Revenge]

[Final Fight Streetwise] [Marvel Vs. Capcom 3] [Marvel Vs. Capcom Infinite] [Mighty Final Fight] [Namco X Capcom] [Ring of Destruction] [Street Fighter III 2nd Impact] [Street Fighter III 3rd Strike]

[Street Fighter Alpha] [Street Fighter Alpha 2] [Street Fighter Alpha 3] [Street Fighter X Tekken] [Street Fighter V] [Super Street Fighter IV] [Ultra Street Fighter IV]

The music for the original SNES port was also used in Final Fight Guy.

Used in:
A Moment of Joy
Any boss beaten excluding Belger
Big Katana Boss
Subway (3rd & 4th parts)
Boil 'em up
This track is not used in the SNES version, more info here.
Burnin' Cycle
Bay Area (3rd part), Up Town (4th part)
Double Andore
West Side (2nd and 3rd parts)
Ending sequence
Both bonus levels
Mad Gear Gang
Nothing Spoiled
Up Town boss beaten
Character select
Slim Subway Smash
Subway (1st part), Up Town (3rd part)
Slum Alleyway
Slum (1st & 3rd parts), West Side (1st part), Up Town (1st part)
Slum Opening
Slums start
Something Putrid by the Bay
Bay Area (1st part)
Street of Luxury
Up Town (2nd part)
Bay Area (2nd part)
Continue Screen
Trashcan Trasher
Slum (2nd part)
Game Over screen

I think I got the titles of these tracks correct, but if anyone thinks otherwise, let me know.

Used in:
Japan (1st part)
Hong Kong (1st and 3rd part), Japan (final boss)
Hong Kong (2nd part), Holland, England (2nd part) Japan (2nd part)
Intro (1st part)
Hong Kong intro
This track isn't used in the game, but it is in the sound test.
Continue screen
Ending sequence
Italy (2nd part), Japan (3rd part)
Character select screen, listen for Ryu's SFII theme in this!
Intro (2nd part)
Japan boss beaten
Italy (1st part)
This track isn't used in the game, but it is in the sound test.
Both bonus levels
Any boss beaten

I'll just list where a track is first played in this game because there are so many levels and routes I can't remember where tracks get re-used.

Used in:
A New Beginning
Used in the endings where Dean says his fight is over.
Chinese Takeout
China Town (4th part)
Down and Out
Main Street (1st part)
Down for the Count
Continue screen
Explosive Situation
Intro (1st part)
For Metro City
Police Station (1st part)
Free and Clear
Any boss beaten, excluding Black
Heavy Hitters
Main Street (Street version boss)
Inner Sanctum
Scrap Yard (4th part)
Law and Disorder
Police Station (2nd part)
Pints and Punks
Main Street (Bar version boss)
RAM Slam!
Break computer bonus game
Short of Breath
Played whenever the game is paused.
Smeared Graffiti
Scrap Yard (1st part)
Smoke Screen
Used in the cutscene after completing the first level.
Straight to the Top
Skull Cross H.Q. (Elevator fight)
Street Fighters
Character select screen
Sunny Skies
Used in the endings where Dean says his fight isn't over.
Wrath of the Skull Cross
Intro (2nd part)