There are 11 Trophies that can be earned in this title.

Bronze

The 0.01 Percent Solution
Got one one ten thousandth of the way through the alternate solution to the Cochlea puzzle

Frequent Tuber
Used all the Bile Tubes in all directions

I'm So Faced
Tried every possible Face-Off feature

Bongo Baby, Yeah
Played the bongos repeatedly

Meet Murray
Introduced Murray to everyone

Silver

The Newlyfed Game
Convinced DeCava that Guybrush is married to Morgan

There is a Brotherhood of Jerks
Accepted into the Brotherhood

La Esponja… Grande?
Acquired La Esponja Grande

Lord of the Locket
Tried every Locket Combination

What a Card
Tried every Tarot Card combination

Gold

Feed the Beast
Defeated DeCava and the Brotherhood

http://www.ps3trophies.org/game/tales-of-monkey-island-chapter-3/trophies/