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

Hero of Lave (Bronze): Arcade - Complete Planet Lave.
Hero of Coventina (Bronze): Arcade - Complete Planet Coventina.
Hero of Nemain (Bronze): Arcade - Complete Planet Nemain.
Hero of Taranis (Bronze): Arcade - Complete Planet Taranis.
Hero of Segomo (Bronze): Arcade - Complete Planet Segomo.
The Tokenizer (Bronze): Arcade - Collect 5 tokens with a single boost.
Scrooge MacBoom (Bronze): Arcade - Get 10 bombs.
Token Hoarder (Silver): Arcade - Collect 15 tokens with a single boost.
Well Prepared (Silver): Arcade - Get 20 bombs.
Shield Blaster (Silver): Arcade - Collect 5 shield tokens while already having a shield.
Scenic Route (Silver): Arcade - Complete a Planet without using boost.
Multiplier Hero (Gold): Arcade - Get 10x multiplier.

Additionally, the following trophies require the bonus downloadable content:

Extra Ships Optional (Bronze): Endless - Survive 7 minutes without dying.
Close Encounters (Bronze): Survivial - Survive 45 seconds without shooting or using bombs.
Late Boomer (Silver): Bomber - Get 15 bombs.
Shock And Awe (Silver): Endless - Destroy 10 nukes.
Brother In Arms (Gold): Split screen Co-op Segomo - Get 3 minutes of continuous Co-Op weapon boost.