Here is the full list of all 19 Picross Fairytale: Legend of the Mermaid achievements.
3 levels in raw without hints
Tutorial finished
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
Solved in less, then 60 seconds
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