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