Successfully complete one of the following tasks to get a trophy:

Sofa Subjugator Bronze (Bronze): Win over five games of Sofa vs. Sofa.
Sofa Subjugator Silver (Silver): Win over 20 games of Sofa vs. Sofa.
Sofa Subjugator Gold (Gold): Win over 50 games of Sofa vs. Sofa.
The Barnard Award Bronze (Bronze): Score over 1500 points in a singleplayer game.
The Barnard Award Silver (Silver): Score over 3000 points in a singleplayer game.
The Barnard Award Gold (Gold): Score over 5000 points in a singleplayer game.
Lonesome and Loving It Bronze (Bronze): Play five singleplayer games.
Lonesome and Loving It Silver (Silver): Play 20 singleplayer games.
Lonesome and Loving It Gold (Gold): Play 50 singleplayer games.
MyBuzz Enthusiast Bronze (Bronze): Play one MyBuzz game.
MyBuzz Enthusiast Silver (Silver): Play 20 MyBuzz games.
MyBuzz Enthusiast Gold (Gold): Play 50 MyBuzz games.
Sofa Surfer Bronze (Bronze): Play one Sofa vs. Sofa game.
Sofa Surfer Silver (Silver): Play 15 Sofa vs. Sofa games.
Sofa Surfer Gold (Gold): Play 40 Sofa vs. Sofa games.
Socialite Bronze (Bronze): Play five multiplayer games.
Socialite Silver (Silver): Play 20 multiplayer games.
Socialite Gold (Gold): Play 50 multiplayer games.
Buzz's New Best Friend (Platinum): All trophies unlocked.

Additionally, there are six secret trophies:

Lightning Reactions (Gold): Correctly answer three questions in a row in under 0.5 seconds.
He's On Fire (Silver): Correctly answer ten questions in a row.
Channel Hopper (Silver): Play through a game on every channel.
Casting Agent (Silver): Play using every contestant.
Flawless Round (Silver): Correctly answer all questions in a multiplayer round with all players.
Amateur Quizwriter (Silver): Create 20 MyBuzz! quizzes.