Here is the full list of all 23 Cat Quest trophies - there are 7 bronze, 7 silver, 8 gold and 1 platinum.
MEOW!
Reached level 99
Cleared all 52 dungeons
Located all monuments
Obtained 66 armor or weapons
Located half the monuments
Completed all 62 side quests
Located one monument
Cleared 25 dungeons
Completed 30 side quests
Obtained 30 armor or weapons
Reached level 50
Obtained all 7 skills
Completed the Main Quest
Defeated the Arcane Dragon
Cleared 5 dungeons
Obtained 4 skills
Defeated the Ice Dragon
Completed 6 side quests
Reached level 10
Defeated the Flame Dragon
Obtained 2 skills
Obtained 3 armor or weapons