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

Amethyst Caverns (Bronze): Complete World 6 in Story Mode.
Argon Refinery (Bronze): Complete World 8 in Story Mode.
Aurora (Bronze): Complete World 2 in Story Mode.
Balls Galore (Bronze): Launch multiple balls into play.
Freon (Bronze): Complete World 5 in Story Mode.
Fringe Benefit (Bronze): Get a score of 10,000,000 in Bonus Mode.
Granule Extractor (Bronze): Complete World 3 in Story Mode.
Kinetic Harvest (Bronze): Complete World 1 in Story Mode.
Krypton Garden (Bronze): Complete World 4 in Story Mode.
Master Batter(Silver): Complete the entire game in a single game session without using a continue.
Neon Mines (Bronze): Complete World 7 in Story Mode.
Storm Warning (Bronze): Trigger a Shard Storm.
The Boss (Silver): Defeat Boss Rush Mode in Under 10 Minutes.
The End of the World (Silver): Complete World 10 in Story Mode.
Xenon Homeworld (Bronze): Complete World 9 in Story Mode.

Additionally, there are three secret trophies:

Schardenfreude (Bronze): Collect 95% of all fragments on any wave.
Overkill (Bronze): Destroy the last brick with a fresh Shard Storm.
Old School (Bronze): Complete a wave without using Blow, Suck, or a Shard Storm.