Here is the full list of all 19 Picross Fairytale: Legend of the Mermaid achievements.
3 levels in raw without hints
Solved in less, then 60 seconds
Atlantis world complete
Mermaid's Cave world complete
Colorful Reef world complete
Sunken Ship world complete
Island world complete
40 levels with 3 stars
Earned 100.000 coins
Tutorial finished
All Trophies collected
50 hints used
20 hints bought
All levels with 3 stars
All siren voices collected
World completed without hints
Mini-game complete
All mini-games complete
100 levels with 3 stars