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

Can't Get Enough (Bronze): Complete 25 different Layouts in 'Infinity' Mode.
Come Out and Play (Bronze): Play 50 Network Games.
Didn't Buy a Clue (Bronze): Complete an entire Tale without using the Hint Button.
Dragon Slayer (Bronze): Complete all Levels in 'Motion' Mode.
I Don't Need No Stinking Power-Ups (Bronze): Complete 'Motion' Mode without using 'Bomb', 'Cannon', 'Ice' and 'Wild number' Power-Ups.
Quick on the Draw (Bronze): Make 5 Quick Matches in-a-row.
Sage (Bronze): Complete all Tales in 'Ancient Tales' Mode.
Score! (Bronze): Score 2500 remaining tiles points in one 'Ancient Tales' level.
Speed Reader (Bronze): Complete an 'Ancient Tales' Layout in less than 2 minutes.
The Duelist (Bronze): Win 5 Network games in-a-row.
The Gunslinger (Bronze): Win 25 Network games.
Why Stop With Just One? (Bronze): Complete a single Tale in 'Ancient Tales' Mode.
Infinity and Beyond (Silver): Complete 100 different Layouts in 'Infinity' Mode.
Matchmaker (Silver): Match 2000 sets of Tiles.
Who's The Master? (Silver): Receive 'Sage', 'Dragon Slayer' and 'Can't Get Enough'.
Who Needs Sleep? (Silver): Complete 'Motion' Mode without losing a single life.

The following trophies require the "Booster Pack" bonus downloadable content:

Be The Best, Mahjong The Rest (Silver): Earn "Who's The Master?" and "Beastmaster" tropies.
Beastmaster (Bronze): Complete all layouts of Animal Pack (Booster Pack - Infinity mode).
Got Layout?! (Bronze): Play through 10 network games on custom layouts.
Mason Apprentice (Bronze): Create 5 Layouts using Layout Editor. Each Layout should have at least 70 tiles in it.
Mason Master (Bronze): Create 10 Layouts using Layout Editor. Each Layout should have at least 100 tiles in it.