Here is the full list of all 39 Muse Dash achievements. It takes around 20-25 hours to unlock all of the achievements on Windows.
Clear a stage
Complete the tutorial stage
Clear 20 stages
Clear 20 Hard Mode stages
Clear 10 Master Mode stages
Achieve grade "S" in a stage
Achieve grade "S" in a Lv.7 or Lv.7+ stage
Get Full Combo in a stage
Get Full Combo in 30 different stages
Reach 300 Combo in a stage
Clear a stage with only one "Great"
Totally knock back 100 BOSS's melee attacks
Select a song through Music Index, Difficulty Filter or Search and play it
Clear a song with Little Devil Marija
Totally knock back 300 Ghosts
Totally collect 300 Red Hearts
Totally collect 500 Notes
Totally knock back 100 enemies or BOSS's remote attacks while playing a Sheet (hold note)
Totally unlock 14 characters
Totally collect 8 elfins
Totally collect 20 illustrations
Play a Lv.11 stage
Clear a Mopemope stage at any difficulty level
Clear a Blackest Luxury Car stage at any difficulty level
Clear a Lights of Muse stage at any difficulty level
Clear a stage of I don’t care about Christmas though at any difficulty level
Clear an umpopoff stage at difficulty level "?"
Finish the tutorial April Fool's Day ver
Play any hidden sheet
Clear a Lv.4 or Lv.4+ stage without collecting Red Hearts
Hit the mascot character on the ENTER screen more than 10 times in a row
Clear a stage without knocking back enemies
Achieve grade "Silver S" in a Lv.7 or Lv.7+ stage
Achieve only 1 "Miss" judgement in a Lv.7 or Lv.7+ stage
Achieve 100% accuracy in a Lv.6 or Lv.6+ stage
Achieve grade "Silver S" of Heart-Pounding Flight on hard mode
Achieve grade "S" of Mujinku-Vacuum Track#ADD8E6 on hard or master mode
Get all achievements except hidden ones
Get all achievements