At the end of every chapter in Balan Wonderworld, you will take on a boss fight to help free a tormented soul. During this time, you will need to take advantage of the costumes you have collected along the way to take down the big baddie and dispel the negativity that has corrupted the person inside.
Upon completing the fight, you will notice that you have a chance to get up to three trophies in the battle by completing what the game calls Fighting Styles. Here is what they are and how to get them.
When the game says Fighting Styles, it is essentially tasking you with damaging the boss in various ways. Each boss will have multiple different attacks that they will hurl at you. What you need to do is use the costumes you have access to in the level to attack them or send their attacks flying back at them to hurt them. If you damage the boss in three different ways, you will be rewarded with all three possible trophies in the fight.
As of this writing, we have not seen any boss fights that require you to go in with any outside costumes. It seems you always will have access to the outfits you need in the level to pull off all three Fighting Styles. You just need to work out the puzzle to get the three different hits landed.