There are 27 Trophies that can be collected.

BRONZE

The day the strength of men failed.
Complete The Battle of the Last Alliance.

It's a dangerous business going out your door.
Complete The Black Rider on the Road.

That is no trinket you carry…
Complete The Camp at Weathertop.

This is no mine… It's a tomb!
Complete Moria: Balin's Tomb.

Let's hunt some Orc!
Complete The Skirmish of Amon Hen.

On the precious…
Complete The Taming of Gollum.

Safe is where I'll keep you.
Complete Track the Hobbits.

You shall not pass!
Complete Moria: Bridge of Khazad-dûm.

The battle for Middle-earth is about to begin…
Complete The Battle of Helm's Deep.

Naughty little fly!
Complete Shelob's Lair.

They have been summoned…
Complete The Paths of the Dead.

You and whose army?
Complete The Battle of the Pelennor Fields.

This day we fight!
Defeat the Mouth of Sauron.

SILVER

Of all the inquisitive Hobbits…
Collect all the Character Profiles.

I think I'm quite ready for another adventure!
Achieve True Adventurer in every level. (Single Player Only)

And away he goes, Precious!
Use Smeagol to defeat Gollum when Gollum is the buddy.

What about second breakfast?
Complete Weathertop in Free Play.

That still only counts as one!
Defeat an Oliphaunt as Legolas.

GOLD

It's gone…
Complete Journey's End.

My precious!
Collect every collectible.

There's some good in this world, Mr. Frodo.
Deliver all the treasure items.

There and back again.
Get 100%. (Single Player Only)

Not with 10,000 men could you do this…
Unlock all characters. (Single Player Only)

Hobbits really are amazing creatures.
Complete all the HUB mini games.

PLATINUM

One Ring to build them all.
Obtain all other Trophies.

2 SECRET

SILVER

One does not simply walk into Mordor…
Simply walk into Mordor. (Single Player Only)

GOLD

The last laugh.
Use Tom Bombadil's laugh to destroy Sauron in Free play.

http://www.ps3trophies.org/game/lego-lord-of-the-rings-vita/trophies/