There are some incredibly obvious ones that don't really have to be pointed out, but for the sake of completion I will anyway. Please say any more you noticed and I'll add them to the list.  Will add more as well. The spoiler tags are just to hide large images. 

Specific game references

• The whole game is reminiscent of River City Ransom
  

•  The level select screen is based on the one found in Super Mario Bros 3

• The green pipes on the map are iconic of SMB in general (obviously)
• Certain Subspace Highway levels (such as the very first one in the game) are copies of the recurring Mario Kart track, Rainbow Road.
  
• The way Scott warps out after defeating a boss is the same as various characters from Mega Man 
• The way Kim flies off on a star after defeating a boss is the same as Kirby.
  

•  The little elf guys that run around on stage 6 that drop stuff if you hit them are based on similar things in Golden Axe
  SPOILER WARNING: Click here to reveal hidden content.

  
  

  

• The bass duel between Scott and Todd Ingram is a parody of Guitar Hero.
• In the final Subspace area, there are floating "Medusa heads" at the end. They are a notoriously annoying enemy from the Castlevania series.
• You can buy an Energy Tank, which are the items in Metroid that raise Samus Aran's maximum HP. The same item appears in Mega Man, which the game parodies more often. However Metroid came out a year before Mega Man so it did technically "invent" the item.
  
• The move that Scott (haven't levelled up other characters yet) has with which he spins through the air, damaging enemies, is a likely reference to Sonic.
• Sonic is also referenced with one of the items - a game called Speedy the Porcupine.
• In stage 4 you can pluck turnips from the ground, carry them around and throw them at enemies, like in Super Mario Bros 2.
• Gideon Graves' logo, the triple G, is designed in such a way that it resembles the Triforce of the Zelda series.
•  The ending is a rip of Mega Man 2's ending, where it's discovered the final boss is actually just a projection by Dr. Wily (or in this case, Gideon) after defeating it and results in the bad guy prostrating himself before the hero (see 00.46 second mark)
  

Possible moveset origins

• The first boss does large, short-range fireballs rather like Sakura's hadouken from the Street Fighter series. Certain characters from the King of Fighters series do the same.

• His "evil hipster chicks" launch air hadouken that are remarkably similar to Akuma's air Gouhadouken. 

•  Scott's special move is a "Vacuum hurricane kick" (Shinkuu Tatsumaki Senpukyaku) from Street Fighter. He also seems to perform a sort of dragon punch in certain combos. Another possible Street Fighter move that Scott has is Chun-Li's headstomp attack. It can even be strung together in the same way.
•  It's also reminiscent of Nightcrawler's headstomp attack in X-Men for the Sega Genesis.
  
• Scott's tech 2 attack (the down, triangle dash move) is reminiscent of Makoto's (Street Fighter Third Strike, SSF4) Hayate special move.
• Scott's tech 1 attack (the up, triangle area attack) is reminiscent of Makoto's Fukiage special move.
•   Ramona's tech attack is similar to Chun Li's Hazan Shuu move (overhead flip kick, see 1:10) 
  
   

General gaming references

•  The whole roast chickens found in stage 4 are the sorts of things you generally get in beat 'em ups that fully heal your health.
• Both the train stage (part of stage 4) and the escalator stage (part of stage 7) are tropes of the beat 'em up genre. Similarly, the first level's night-time streets with breakable phone booths and parking metres is an almost ubiquitous setting in the genre.
  

• Survival Horror, the game's survival mode, is of course a reference to the (usually) zombie-infested genre of the same name.
• The giant robot boss of stage 6 is designed in a way that a lot of giant bosses in the 8- and 16-bit era were; body in the background, two giant hands as separate sprites moving around all over the place.

Non-gaming references

• The large robot boss is revealed in the end credits to be called "Kazinger K", an obvious play on " Mazinger Z", a famous anime mech.

I was hoping somebody would start this up. =) 
 
Here's a few off the top of my head: 

• The ending is a rip of Mega Man 2's ending, where it's discovered the final boss is actually just a projection by Dr. Wily (or in this case, Gideon) after defeating it and results in the bad guy prostrating himself before the hero (see 00.46 second mark)
• Scott's tech 2 attack (the down, triangle dash move) is reminiscent of Makoto's (Street Fighter Third Strike, SSF4) Hayate special move.
• Scott's tech 1 attack (the up, triangle area attack) is reminiscent of Makoto's Fukiage special move.
• Scott's headstomp attack (triangle, while in air) is reminiscent of Nightcrawler's headstomp attack in X-Men for the Sega Genesis.
• The bass duel between Scott and Todd Ingram is a parody of Guitar Hero.
• This one may be more of a stretch, but the Super Fighting Robot boss battle is reminiscent of both Sonic & Knuckles' Phase 1 Final Boss battle (see 1.55 time mark) and Legend of Zelda: Ocarina of Time's Bongo Bongo boss battle.
• An even bigger stretch, but the music for Nega Scott Appears sounds like a sped up version of Pokemon's Lavender Town music.

Awesome.

The first subspace highway is obviously a nod to Mario's Rainbow Road levels.
NegaScott seems very much like Shadow Link.

Cool!  Had the same observations as well on most of the list.  Some additional ones: 
 
- Ramona's tech attack is similar to Chun Li's Hazan Shuu move (overhead flip kick) 
- Scott's hurricane kick move is also stationary, similar to Double Dragon's hurricane kick move 
- The last stage before fighting true Gideon is reminiscent of the Teenage Mutant Ninja Turtles: Turtles in Time technodrome stage (including the robot enemies) 
- Ramona's ex-boyfriend twins are like Double Dragon's Billy and Jimmy

This isn't a videogame reference but cool none the less: The director who comes out every now and then during the Lucas Lee level is totally supposed to be Edgar Wright. It looks exactly like him. 

• This one may be more of a stretch, but the Super Fighting Robot boss battle is reminiscent of both Sonic & Knuckles' Phase 1 Final Boss battle (see 1.55 time mark) and Legend of Zelda: Ocarina of Time's Bongo Bongo boss battle.

Thanks for the extra references guys. I've left some of them out because either they were too general (like NegaScott being like Dark Link - an evil version of the hero is a very common trope in fiction) or because I haven't seen them for myself. In the case of the latter, I'll check some of them out (watching a video of the Technodrome stage from Turtles in Time now).

On the Energy Tank, it is definitely a Mega Man energy tank, not a Metroid one.  It is a consumable item that restores health, and is carryable as an item.  It even uses a similar sprite to the Mega Man energy tank when you take it to go.
 
Scott's Level 14 attack, Grand Slam is named after and used like the Grand Slam attack from River City Ransom, which also allows you to swing your weapon much faster, and 3 times per button press.  The Spin attack could also be a nod to RCR's Acro Circus attack. 
 
The character select screen is designed based on the US version of SMB 2
 
Also, sadly, performing 8 head-stomps in a row did not result in a 1-up.  Missed opportunity!
 
Edit: More stuff!
 
The enterance to the secret shop in level 6 is very simlar to the houses in Zelda 2: The Adventure of Link.  The shop it's self looks like where you meet the wizards in Zelda 2 to learn new spells, and this is also where you can buy each character's secret techniques!
 
While paused, the game plays a repeating bass line, slightly similar to that of Battletoads.
 
The code to enable blood (ABAXABB in the XBox 360 version), is a refernce to the code to enable blood in the Genesis version of Mortal Kombat (ABACABB)

Well, that spoiled the ending for me.