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