SONYA BLADE:
Pink: You and Your Hand or Pussycat Dolls: I Don't Need A Man (watch the intro and you'll understand...)
TANYA:
Rihanna: Unfaithful
To hear the songs:
You and Your Hand - http://youtube.com/watch?v=xbll0nc_cv4 (real film clip)
I Don't Need A Man - http://youtube.com/watch?v=kKFyUjvwqcI (real film clip)
Unfaithful - http://youtube.com/watch?v=pyX21c8bxUM (real film clip)

jax- ain't nothin but a g thang
noob saibot- enter sandman, i don't know why, i just think it fits him
blaze-it's the end of the world as we know it
scorpion- Unforgiven II by metallica

My apologies for having just one character in the initial post.
I just have to look into them more.
But for now.....Sub-Zero - Vanilla Ice - Ice Ice baby lol

Ok...............
SHANG TSUNG:------------Unholy performed by Kiss
CYRAX or SEKTOR:---------War Machine performed by Kiss

Here are some more:
KHAMELEON:
Kelly Clarkson - Gone
(http://youtube.com/watch?v=0Nre78nFMIo - not the film clip, but that is the song)
ASHRAH:
Tina Turner - Simply The Best
(http://youtube.com/watch?v=R1tCrfmDquc - not the film clip, but that is the song)
RAIDEN:
Delta Goodrem - Electric Storm
(http://youtube.com/watch?v=9QKNMEFn-is - not the film clip, but that is the song)

Shao Kahn's song is
SHUT YOUR MOUTH: by Pain I'll take it one step further and link the video http://www.youtube.com/watch?v=Cqd5nvoN-Kg

BLAZE-ELEVATION BY U2

Sektor:
Obsolete - Fear Factory

Ermac - We come One (Faithless)

Blaze:
Alkaline trio - Armageddon

For the heck of it:
Blaze - Stronger by Britney Spears
Motaro - What's left of me by Nick Lachey
Chameleon - Ridin' by Chamillionaire
Sareena - Permiscuous Girl by Nelly Furtado
Bo Rai Cho - Bounce by Fatty Koo
Sheeva - She's a lady by Tom Jones
Reptile - Invisible by Clay Aiken
Frost - Ice Queen by Within Temptation

Lol, the two best by far

endless darkness- Noob saibot
Bring me to life- Liu Kang
Rain- Rain
Die motherfucker Die- Scorpion
Quan chi- how can i live

Hsu Hao : Slipknot - I am hated
Sektor : KoRn - Twizted Transistor or Fear factory - Hi-tech hate
or the Ketchup Song

scorpion - "scorpion" by megadeth
bo rai cho - "Fat" by weird al yankovic
hotaru - "asshole" by denis leary
khameleon OR chameleon - "karma chameleon" by i forgot....
cyrax - "all about the pentiums" by Weird al yankovic....
Rain - "Purple Rain" by Prince (obviously)

drahmin : releasing the deamons by godsmack
baraka : edge crusher by fearfactory

Johnny Cage =
Rob Zombie_ Never gonna stop
(See him in the versus screen!)
Frost =
R.H.C.P._ The zephir song
Sub-zero =
Mentos theme
Khameleon =
They Might Be Giants- Your not the Boss of Me Now
(Are you in the game? Khameleon answer= Yes, no, maybe I don't know...)
Stryker =
Benny Hill theme

sektor and cyrax a song jedi fight in star wars mmm maybe obiwan vs anakin

Chameleon
Karma Chameleon - Culture Club
Moloch
Duran Duran - Hungry like the Wolf

Fujin = Scorpions - Rock you like hurricane
Sindel = Queen - Killer Queen
Liu Kang = Queen - We Are The Champions , The Misfits - Astro Zombies or The Cranberries - Zombie
Stryker = Dennis Leary - Im An Asshole
Noob or Scorpion = ACDC - Highway To Hell
Bo' Rai Cho = Metallica - Whiskey In The Jar

+ Stryker = Static Addict - Lil Wyte (one of my fav song's.)
+ Fujin = Immortal - Skinny Puppy (another one of my fav song's.)
+ Sareena = Your Body - Pretty Ricky
+ Kobra = Keep it Kali - Kottonmouth Kings
+ Sindel = One of a kind - Atmosphere
+ Darrius = Dope boy magic - Yung Joc
+ Smoke = 2nd Hand Smoke - TWiZTiD
+ Sheeva = Im bossy - Kelis
+ Liu Kang/Kung Lao = Someday - Flipsyde
+ Nitara = Sun Dosen't rise at all - Mushroomhead
+ Havik = Loco - Coal Chamber
+ Mileena/Kitana = Girlfight - Brooke Valentine
I got more but i'll stop here.

Niiiice
I will also add:
BARAKA or MILEENA or NITARA - Nelly Furtado: Maneater
KINTARO - Survivor: Eye of the Tiger
KENSHI - Lifehouse: Blind
KUNG LAO or JADE - Ashlee Simpson: Shadow

kabal = the nomad - iron maiden
sub-zero= foreigner - cold as ice
stryker = bad boys - bob marley
scorpion = drop it like its hot - snoop dogg
cyrax = mean mr. mustard - the beatles
sektor = The Ketchup Song - las ketchup
smoke = smoke on the water - deep purple
fujin = Colors Of The Wind - Vanessa Williams
raiden = electric man - mansun
shao kahn = i'm bad - micheal jackson
mileena = she's a maniac - flashdance
darrius = when im nothing - dmx
mokap = christmas lights - donna lewis

oh..
Tanya = Gimme That - Chris Brown / Lil Wayne