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
Completed the Main Quest
Completed all 62 side quests
Cleared all 52 dungeons
Obtained 66 armor or weapons
Reached level 50
Cleared 25 dungeons
Defeated the Arcane Dragon
Completed 30 side quests
Defeated the Ice Dragon
Obtained 30 armor or weapons
Obtained all 7 skills
Located all monuments
Obtained 4 skills
Reached level 10
Defeated the Flame Dragon
Located half the monuments
Completed 6 side quests
Cleared 5 dungeons
Obtained 2 skills
Obtained 3 armor or weapons
Located one monument