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