There are 47 Trophies that can be collected.

Bronze

The Best There Is    
Beat Arcade mode on the hardest difficulty.     

Saving My Quarters    
Beat Arcade mode without using any continues.     

The Points Do Matter    
Earn 400,000 points in Arcade mode.

Missions? Possible.    
Clear 120 missions in Mission mode.

Above Average Joe    
Land a Viewtiful Combo. (Arcade/online only)     

Mutant Master    
Land an Uncanny Combo. (Arcade/online only)     

Mega Buster    
Use 1,000 Hyper Combo Gauge bars. (Arcade/online only)     

Defender    
Block 100 times. (Arcade/Online only)     

Advancing Guardian    
Perform 100 Advancing Guards. (Arcade/Online only)     

A Friend in Need    
Perform 100 Crossover Assists. (Arcade/Online only)     

First Strike    
Land 50 First Attacks. (Arcade/Online only)     

Savage Playing    
Perform 50 Snap Backs. (Arcade/Online only)     

Quick Change-Up    
Perform 50 Crossover Counters. (Arcade/Online only)     

Perfect X-ample    
Use X-Factor 50 times. (Arcade/Online only)     

Gravity? Please…    
Land 50 Team Aerial Combos. (Arcade/Online only)     

Mighty Teamwork    
Land 30 Team Aerial Counters. (Arcade/Online only)     

Big Bang Theory    
Perform 30 Hyper Combo Finishes. (Arcade/online only)     

Hard Corps    
Perform 30 Crossover Combination Finishes. (Arcade/online only)     

Training Montage    
Play in Offline Mode for over 5 hours.

Seductive Embrace    
Play in Online Mode for over 5 hours.

Brave New World    
Participate in any mode online.     

Hellbent    
Participate in 100 matches online.     

Dreaded Opponent    
Participate in 200 matches online.

Incredible    
Win without calling your partners or switching out in an online match.     

Need a Healing Factor    
Win without blocking in an online match.     

Crazy Good    
Surpass the rank of Fighter.

Hotshot    
Win 10 battles in Ranked Match.     

Slam Master    
Win 50 battles in Ranked Match.

Noble Effort    
Get a 5-game win streak in Ranked Match.     

Assemble!    
Participate in an 8 player Lobby online.

A Warrior Born    
Earn 5,000 Player Points (PP).     

Devil with a Blue Coat    
Earn 30,000 Player Points (PP).

Passport to Beatdown Country    
Fight in all of the stages.

Silver

High-Score Hero    
Earn 500,000 points in Arcade mode.

Up To The Challenge    
Clear 240 missions in Mission mode.     

Master of Tasks    
Clear 480 missions in Mission mode.

Training in Isolation    
Play in Offline Mode for over 30 hours.

Rivals Welcome    
Play in Online Mode for over 30 hours.

Forged From Steel    
Participate in 300 matches online.     

Full Roster    
Battle against all characters online.

Mega Good    
Surpass the 6th rank of any class.

Fighting Machine    
Win 100 battles in Ranked Match.

Dominator    
Collect 100 titles.

Divine Brawler    
Earn 100,000 Player Points (PP).

Gold

Waiting for the Trade    
View all endings in Arcade mode.

Comic Collector    
Unlock all items in the Gallery.

Platinum

The Ultimate    
Obtain all Trophies.

http://www.ps3trophies.org/game/ultimate-marvel-vs-capcom-3-vita/trophies/