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