There are 11 Trophies that can be collected.

Bronze

Nerves of Steel
Get your characteristic in Nervous System section

Brain Maniac
Get your characteristic in Intelligence section

Nice Start
Complete any test

Quick Learner
Improve any test's result 10 times in a row throughout the day (In single player training mode)

Silver

True Friend
Describe Yourself In social tests so that your assessment matches your friends opinion (again probably requiring padding)

Self-Known
Request At least 10 assessments from your friends in Social Test

Racer
Outlast at least for 20 seconds in Brownian single player game

Solved In 300 Seconds
Complete the Intelligence Test in 300 seconds without mistakes

Gold

Psychologist
Assess 10 friends in Social Tests (sounds like it might have to be padded)

2 Secret

http://www.ps3trophies.com/forums/trophy-news/55154-test-yourself-psychology-trophy-list.html