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