Here is the full list of all 13 Bombing Busters trophies - there are 11 bronze, 1 silver and 1 gold.
Defeat the Spark Boss
Defeat all Bosses with 3 stars
Complete all 30 levels with 3 stars
Defeat the Fire Boss
Complete 20 levels with 3 stars
Defeat the Wind Boss
Defeat the Ice Boss
Complete 10 levels with 3 stars
Defeat the Fairy Boss
Complete all Fairy levels before the Fairy Boss
Complete 5 levels with 3 stars
Complete the first level
Complete a level with 3 stars