Here is the full list of all 36 Patapon Remastered trophies - there are 13 bronze, 19 silver, 3 gold and 1 platinum.
Unlock all trophies.
Disrupt the beat 15 times in a single level.
Defeat Gorl.
Acquire 9 Divine items.
Defeat Shooshookle.
Defeat Shookle.
Clear "Facing Gate Ghoul Baban".
Defeat a level 10 boss.
Lose a boss battle with 5% or less of the boss' health remaining.
Defeat Dogaeen.
Use each type of unit at least once.
Create a Barsala Ultra Rarepon.
Complete a mission with just one remaining Patapon (in addition to the Hatapon).
Defeat Ciokina.
Defeat Cioking.
Unlock Ton Kampon's mini game.
Defeat a boss with an army of only one unit type.
Clear "Meden Kidnapped".
Create a Mogyoon Ultra Rarepon.
Defeat Dokaknel.
Defeat Gaeen.
Clear "Battle! Mt. Bachikoi Fort".
Unlock the Rah Gashapon's mini game.
Clear "Convoy Escort".
Complete 25 missions with a single unit without being reborn.
Collect 300 ka-ching in one mission.
Unlock Kon Kimpon's mini game.
Defeat Zaknel.
Unlock Fah Zakpon's mini game.
Summon the power of JuJu to perform a miracle.
Defeat Majidonga.
Defeat Dodonga.
Clear "Gong the Hawkeye".
Remain in Fever status for one minute.
Unlock Pan the Pakapon's mini game.
Learn the basics by completing the first mission.