Here is the full list of all 77 Poly Island achievements.
Finish level 1.
Finish level 5.
Finish level 10.
Finish level 15.
Finish level 20.
Finish level 25.
Finish lvl 30.
Finish lvl 35.
Finish lvl 40.
Finish lvl 45.
Finish lvl 50.
Finish a level without dying.
Finish 2 levels in a row without dying.
Finish 3 levels in a row without dying.
Finish 4 levels in a row without dying.
Finish 5 levels in a row without dying.
Get killed by lasers.
Get killed by spikes.
Get killed by a rolling ball.
Get killed by a bullet.
Get killed by a crushing pillar.
Finish the game.
Finish 2 levels without dying.
Finish 3 levels without dying.
Finish 4 levels without dying.
Finish 5 levels without dying.
Finish 6 levels without dying.
Finish 7 levels without dying.
Finish 8 levels without dying.
Finish 9 levels without dying.
Finish 10 levels without dying.
Finish all the levels on "max lives" mode.
Finish a level on "max lives" mode.
Finish 5 levels on "max lives" mode.
Finish 10 levels on "max lives" mode.
Finish 20 levels on "max lives" mode.
Finish 30 levels on "max lives" mode.
Finish 40 levels on "max lives" mode.
Finish all the levels on "no checkpoint" mode
Finish a level on "no checkpoint" mode.
Finish 5 levels on "no checkpoint" mode.
Finish 10 levels on "no checkpoint" mode.
Finish 20 levels on "no checkpoint" mode.
Finish 30 levels on "no checkpoint" mode.
Finish 40 levels on "no checkpoint" mode.
Get killed by lasers 5 times.
Get killed by lasers 10 times.
Get killed by lasers 15 times.
Get killed by lasers 20 times.
Get killed by spikes 5 times.
Get killed by spikes 10 times.
Get killed by spikes 15 times.
Get killed by spikes 20 times.
Get killed by a rolling ball 5 times.
Get killed by a rolling ball 10 times.
Get killed by a rolling ball 15 times.
Get killed by a rolling ball 20 times.
Get killed by a bullet 5 times.
Get killed by a bullet 10 times.
Get killed by a bullet 15 times.
Get killed by a bullet 20 times.
Finish a level on "time challenge mode".
Finish all levels on "time challenge mode".
Finish 5 levels on "time challenge mode".
Finish 10 levels on "time challenge mode".
Finish 20 levels on "time challenge mode".
Finish 30 levels on "time challenge mode".
Finish 40 levels on "time challenge mode".
Finish the game with time, life and checkpoint mod enabled.
Finish 5 levels with time, life and checkpoint mod enabled.
Finish 10 levels with time, life and checkpoint mod enabled.
Finish 15 levels with time, life and checkpoint mod enabled.
Finish 20 levels with time, life and checkpoint mod enabled.
Get killed by crushing pillars 5 times.
Get killed by crushing pillars 10 times.
Get killed by crushing pillars 15 times.
Get killed by crushing pillars 20 times.