Hack and Slash: God of War III
FPS: Bioshock Infinite
TPS: Gears of War 3
Racing: Forza 4 
RPG: Skyrim 
Action-RPG: Mass Effect 2
Platformer: LittleBigPlanet 2
Stealth Shooter: Deus Ex: Human Revolution 
Sandbox: Red Dead Redemption 
Survival Horror: The Last of Us

Some of the games you mentioned, like BioShock and PGR4, were released before 2010.

Let's see...

FPS - Halo: Reach

3rd Person Shooter - Kid Icarus: Uprising

(favorite genre) 3D Platformer - Sonic Generations

2D Platformer - Rayman Legends

RPG - Pokemon HeartGold if that counts (released in 2009 in Japan, 2010 everywhere else), Kingdom Hearts 3D otherwise

Racing - Need for Speed: Hot Pursuit

Action-Adventure - Zelda: Wind Waker HD if that counts (HD remake of favorite game), Zelda: Link Between Worlds otherwise

Sandbox - Assassin's Creed IV: Black Flag

Strategy - Fire Emblem Awakening

Arcade - Pac-Man Championship Edition DX

Fighting - Super Smash Bros for 3DS (likely to be replaced by Smash Wii U upon launch), and if you're one of THOSE assholes who doesn't count Smash as a fighting game, then Injustice

Survival horror - Alan Wake

Party - Nintendo Land

Yea... I think that covers the genres I play. If it can count, you can throw in Bayonetta for hack and slash (JP 2009, elsewhere 2010), otherwise I got nothin. Thoughts?

I haven't played many of the games you listed. Though among the games I have played, like AC IV: BF, Alan Wake and Bayonetta, I really liked all the games. Though none of them are my favourites in the genre, they are all really good games. As you can see from my list, I'll choose GTA V, The Last Of Us and God Of War III over these games in their respective genres.

Fair enough, I just got a PS3 recently to close out the 7th gen (and Kingdom Hearts 1.5 and 2.5...), so I haven't played Last of Us, Uncharted 3, God of War III, or Gran Tursimo 6. Grand Theft Auto and Metal Gear just isn't to my taste sadly. I was seriously considering Bioshock Infinite for favorite FPS before remembering my playtime with friends in Reach (in the vicinity of 900 hours if memory serves). While I got Most Wanted for my Wii U recently, I haven't really played it enough to fairly place against Hot Pursuit. And if I had put WRPG, Skyrim would have to be it. 

So you just got a PS3? That explains a lot, I think the following games are must haves. They are pretty cheap right now, and you should definitely get them.

1. Uncharted trilogy

2. The Last of Us

3. God Of War: Saga (It includes 5 God Of War games, all of which are must-plays)

If you like the God Of War games included in Saga, you should also get God Of War: Ascension. Your favourite TPS and hack n slash game might change after playing these games

Oh shit loool, from reading the title to posting I completely forgot the rule, which is weird for me...  I'm going senile :(

Will edit.