I'm gonna see how many references I actually understood, and list them here. Feel free to add all the ones that I missed! Here we go in no particular order (not counting straight cameos and the many references to internet slang):

Movies, TV, magazines and music
A-team
Teenage Mutant Ninja Turtles
Batman
Star Warts
Vanilla Ice
Saved By The Bell
Soylent Green
Duck Tales (or Scrooge McDuck in general)
Dukes of Hazzard
Die Hard
Back To The Future
Bill & Ted
Ghostbusters
Robocop
General Hospital
Scooby Doo
Knight Rider
The Running Man
Terminator
Michael Jackson
World Wrestling Federation
Blues Brothers
He-Man
ZZ Top
Napoleon Dynamite
Devo
Mötley Crue
Friday the 13th
Fabio Lanzoni
Elvis Presley
Fresh Prince of Bel Air
Ghost Dad
The Simpsons
Thundercats
Daft Punk
Family Matters
MC Hammer
Married with Children
Star Trek
Ernest movies
MacGyver
Tenacious D
Nintendo Power
Spiderman
E.T.
Last Action Hero
Magnum P.I.

Video games
Bomberman
Mega Man
Punch-Out
Duke Nukem Forever
Metal Gear
Contra
Sonic The Hedgehog
Mario Bros
Super Mario Bros
Zero Wing
Secret of Monkey Island
Halo
Bionic Commando
Oregon Trail
Rad Racer
Donkey Kong
Skate Or Die
Frogger
Pitfall
Mortal Kombat
Legend Of Zelda
Smash TV
Space Quest
Grand Theft Auto (duh)
Excitebike
Splosion Man
Earthworm Jim
Tapper
Ikari Warriors
Ninja Gaiden
Paper Boy
Duck Hunt
Rygar
Dr. Mario
Street Fighter
Double Dragon
Bayou Billy
Ghosts 'n' Goblins
Blaster Master
Pro Wrestling
Ironsword: Wizards & Warriors II
Sinistar
Gargoyle's Quest
X-COM
Jackal
Leisure Suit Larry
Doki Doki Panic
Scrap Metal
Little Nemo: The Dream Master
Pac-Man
Ms. Pac-Man
Urban Champion
Q*bert
The Incredible Machine

Companies and stores
Taco Bell
Acclaim
Apogee
Apple
Starbucks
Toys 'R' Us
Subway
T.G.I. Sunday's
Data East
Pizza Hut
IBM
U-Haul
Little Caesars

I'm kinda disappointed because I didn't see any E.T or MacGyver references. Imagine: an E.T-ish creature sitting in the front basket of a bike, and pressing shift would make him shoot a laser from his finger (or alternatively, make the bike jump or hover for a short distance).

Good call, I added Duck Hunt among with Rygar, Double Dragon, Street Fighter, Dr. Mario, Bayou Billy, Friday the 13th, Devo, Mötley Crue, Elvis Presley and Fabio.
Also, I did mention Grand Theft Auto already, and the second game in particular because the Spray 4 Pay shops are a reference to the Pay 'n' Spray shops (I can't remember if they were called that in the later games too).

I said in the first post that I didn't count straight cameo appearences, so I didn't mention the appearences by John Romero, Phil Fish, Epic Meal Time and so on.

Also, I should have mentioned that when I listed Super Mario Bros and Legend Of Zelda, I meant that the game references those two series in general, because I didn't want to list every individual title.

I did beat all the main story missions and the side missions too, but I don't remember seeing those references. I guess I gotta go find them now.

I haven't seen Ghost Dad, but I added it to the list anyways. I also added Fresh Prince of Bel Air and Ghosts 'n' Goblins.
Wow. 71 movie, TV, music and video game references. Maybe I should add the company references in there too.

Updated the list with a bunch more references! Added The Simpsons, Blaster Master, Pro Wrestling, Taco Bell, Starbucks, Toys 'R' Us, Apogee, Acclaim, Subway and T.G.I. Sunday's.
81 references so far.

Added Thundercats and Data East to the list. 83 references so far. I think the game is starting to run out of references by this point.

Added Daft Punk and Ironsword: Wizards & Warriors II to the list.

Added Family Matters, MC Hammer, Married with Children and Pizza Hut to the list. Geez, how many more references can this game possibly have?!

Added Star Trek, Ernest, Sinistar and IBM to the list. 93 references! Just.... wow.