There are 43 Trophies that can be collected.

Bronze

First Design    
Cleared a battlefield created using Musou Battlefields.    

First Forge    
Upgraded a weapon using Weapon Fusion.    

First Purchase    
Purchased a weapon.    

Joining Hands    
Cleared a stage using either online play or 2-player mode.    

Reaching Beyond    
Purchased a weapon using crystals.    

The Mead of Life    
Held a party.    

Firm Bonds    
Raised the bonds of one pair of officers to its maximum.    

Experienced Controller    
Used all mountable siege weapons.    

Musou Fighter    
Achieved 1000 K.O.s.    

Blinding Speed    
Cleared a stage in under 5 minutes.    

Dance of the Blade    
Successfully performed a combo of 1000 or more.    

Overwhelming Victory    
Obtained over 10,000 EXP in a single battle.    

Blinding Light    
Used a True Triple Attack.    

Challenger    
Successfully completed a mission.    

Perfect Partner    
Raised compatibility of a weapon to maximum.    

Grand Purchase    
Purchased a weapon that cost over 10,000 gems.

Silver

True Musou    
Obtained all trophies.    

Musou Collector    
Obtained level 4 weapons for all officers.   

Gallery of Heroes    
Unlocked all officers for play.    

Master of Time    
Cleared all scenarios.    

Musou Dilettante    
Obtained every equippable item.    

Slave of Sound    
Listened to all the music.    

Veteran Warrior    
Entered battle over 100 times with any one officer.    

A Kaleidoscope of Time    
Unlocked all wallpaper.    

The Dandy Captain    
Unlocked all costumes.

Platinum

True Musou    
Obtained all trophies.

18 Secret

Bronze

The Flame of Hope    
Cleared Chapter 1 in Story Mode.    

A Hidden Smile    
Cleared Chapter 2 in Story Mode.    

Unwavering Hope    
Cleared Chapter 3 in Story Mode.    

Changing the Past    
Cleared all 10 Redux stages.    

Banner of Victory    
Defeated the Hydra.    

Traversing Time    
Returned to the past using Kaguya's power.

Silver

In a Broken World    
Cleared Story Mode with NORMAL ending.    

Land of Peace    
Cleared Story Mode with GOOD ending.    

The Fated Place    
Cleared Story Mode with TRUE ending.    

A Warrior of Wealth    
Gathered over 500,000 gems.    

Noble Warrior    
Gathered over 100 crystals.    

Shopping Expert    
Purchased over 100 weapons.    

Otherworld Extravagance    
Held over 100 parties.    

Master of Battle    
Cleared any stage without taking any damage.    

Musou Warrior    
Achieved 3000 K.O.s.    

Mission Master    
Completed more than 10 missions in one battle.

Gold

Peak of Performance    
Cleared all stages at Chaos difficulty.    
 
Musou Millionaire    
Gathered over 900,000 gems.

http://www.ps3trophies.org/game/warriors-orochi-3/trophies/