There are 12 Trophies that can be collected.

Bronze

Your Tax Dollars at Work   
Submitted a successful puzzle answer on the first try while in the field.

'Yeah, I'm into some stuff.'   
Helped Steve with his shipping problems.

'The Cold Will Creep Right Up On Ya.'   
Talked to Scruffman after seeing the unseeable outside of the diner.

Hot Dish   
Check into Valda's Inn

Crossword Double Cross   
Thwarted Sheriff Bahg.

Hvor er Olav Welhavven?   
Took a transdimensional space voyage.

'Checking Out.'   
Overcame your fear of Scoggin' hidden people and stay on the case.

Silver

Fidelity, Bravery, Integrity and Madness   
Completed the case and saw into the maw of madness.

Thorough Investigation   
Complete every puzzle in Scoggins.

Kicked the Habit   
Completed the case without chewing a piece of gum.

The Professional   
Closed the case with the rank of 'Puzzle Agent.'

Gold

Perfect Puzzle Agent   
Agent, you never submitted a wrong answer.

http://www.ps3trophies.org/game/nelson-tethers-puzzle-agent/trophies/