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