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
Obtained 66 armor or weapons
Reached level 99
Completed the Main Quest
Completed all 62 side quests
Obtained all 7 skills
Located all monuments
Defeated the Arcane Dragon
Cleared 25 dungeons
Reached level 50
Defeated the Ice Dragon
Defeated the Flame Dragon
Obtained 30 armor or weapons
Located half the monuments
Obtained 4 skills
Completed 30 side quests
Cleared 5 dungeons
Completed 6 side quests
Reached level 10
Obtained 3 armor or weapons
Obtained 2 skills
Located one monument