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