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