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