There are 16 Trophies that can be collected.

BRONZE

Starting Sparks
Got a gold on all the Fire Pit levels.

Lumber Jack
Got a gold on all the Wood Pile levels.

Tying It Up
Got a gold on all the Dock levels.

Royal Flush
Got a gold on all the Outhouse levels.

Wheeling & Dealing
Got a gold on all the Wagon Wheel levels.

Shedarific
Got a gold on all the Shed levels.

Cabin Fever
Got a gold on all the Cottage levels.

Shatter the Sky
Got a gold on all the Satellite levels.

Rainbow
Got 4 or more flames all of different colors at once.

Dead Ant
Eliminated 5 Ants.

Beetle Bopper
Eliminated 10 Beetles.

Web Wizard
Eliminated 20 Spiders.

Bomb Blaster
Eliminated 20 Fireflies.

Electrician
Eliminated 30 Electric Bugs.

Soaking Wet
Eliminated 30 Water Bugs.

GOLD

Master of the Flame
Got a gold on every level.

http://www.ps3trophies.com/forums/playstation-trophy-news/23686-burn-rope-playstation-vita-trophy-list.html