12 Bronze
1 Silver
1 Gold

Bronze

10 times Wild Draw Four
When you use the Wild Draw Four Card 10 times in tournament.

2 times Wild card in a game
When you use the Wild Card 2 times in a round in tournament.

Challenge UNO
When you successfully catch someone forgetting to call UNO in tournament.

Challenge Wild Draw Four
When you successfully challenge an illegal use of a Wild Draw Four Card in tournament.

Finish 10 games
When you complete 10 rounds (win or lose) in tournament.

First UNO
When you first call UNO in tournament.

First win
When you win a round for the first time in tournament.

Invite your friend
First time you invite your friend to play with you.

Play 50 games online
Play 50 online UNO rounds.

Play online
First time you play a round online.

Win 10 games online
Win 10 online UNO rounds.

Win 10 matches
When you win a total of 10 matches in tournament.

Silver

Win game without drawing cards
When you win a round without drawing any cards in tournament.

Gold

Top 50 in the world
Check the leaderboard when in the TOP 50.

Source: http://ps3trophyguide.com/game/view/707