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