Here is the full list of all 27 Just Shapes & Beats trophies - there are 8 bronze, 12 silver, 6 gold and 1 platinum.
Obtain 15 Rank S
Obtain Rank S without dashing (Challenge mode)
Complete a challenge with Rank A or higher on all levels
Obtain 20 Rank A
Complete a playlist of all Story mode bosses
Obtain 10 Rank A
Complete a playlist of 10 songs or more
Get 2000 Beat Points
Obtain 10 Rank B
Dash 2500 times (Challenge mode)
Rescue 50 friends (Challenge mode)
Dash 1000 times (Challenge mode)
Complete a level without dashing (Challenge mode)
Complete Story
Beat 'Final Boss' (Story mode)
Beat 'Close To Me' Boss (Story mode)
Beat 'Lycanthropy' Boss (Story mode)
Beat 'Barracuda' Boss (Story mode)
Beat 'New Game' Boss (Story mode)
Beat tutorial Boss (Story mode)
Collect all Just Shapes & Beats trophies
Rescue 250 friends (Challenge mode)
Complete 10 challenges
Complete 25 challenges
Complete 50 challenges
Get 7500 Beat Points
Get 15000 Beat Points