There are 51 Trophies that can be collected.

Bronze

Dragon Sword Master
Defeat 1000 enemies with the Dragon Sword

Vigoorian Flail Master
Defeat 1000 enemies with the Vigoorian Flail

Dabilahro Master
Defeat 1000 enemies with the Dabilahro

War Hammer Master
Defeat 1000 enemies with the War Hammer (Hayabusa)

Kitetsu Master
Defeat 1000 enemies with the Kitetsu

Dual Kitana Master
Defeat 1000 enemies with the Dual Kitana

Lunar Staff Master
Defeat 1000 enemies with the Lunar Staff

Wooden Sword/Unlabored Flawlessness Master
Defeat 1000 enemies with the Wooden Sword/Unlabored Flawlessness

Projectile Weapons
Defeat 100 enemies with projectile weapons

Feat of Ultimate Destruction
Use the strongest Ultimate Technique on 100 enemies

Izuna Drop Master
Use the Izuna Dro on 100 enemies

Guillotine Throw Master
Use the Guillotine Throw on 100 enemies

Fire Wheels Master
Use Fire Wheels successfully 100 times

Inferno Master
Use Inferno successfully 100 times

Ice Storm Master
Use Ice Storm successfully 100 times

Inazuma Master
Use Inazuma successfully 100 times

Cleared The Way of the Ninja
Completed Chapter 1

Cleared 2 Trials
Clear all Ninja Tutorials and Path to the Ultimate Ninja.

Platinum

True Ultimate Ninja
Collect all trophies

32 Secret

http://www.ps3trophies.com/forums/playstation-trophy-news/12450-ninja-gaiden-sigma-plus-trophy-list.html