Here is the full list of all 12 Rally Copters trophies - there are 8 bronze, 3 silver and 1 gold.
Complete all phase 3 levels with the Mosquito
Unlock all helicopters
Complete all levels
Get a platinum medal
Complete a flag run without taking damage
Perform a successful flip-dodge
Travel 100 meters in a single dive
Earn a gold-plus (or platinum-plus) medal
Complete a phase-3 level with no guideline in one try
Reach 50 tries in a level
Crash your helicopter after crossing a finish line
Obtain a full landing bonus