Here is the full list of all 40 Kubinashi Recollection achievements.
Clear Dream Fields through stage 5
Clear Lost Bamboo Thicket through stage 5
Clear Azure Winter through stage 5
Clear Magic Forest through stage 5
Clear Ultramarine Rain through stage 5
Clear Outside World through stage 5
Clear Scarlet Devil Mansion Interior through stage 5
Clear Shining Needle Castle through stage 5
Clear Dream Scraps through stage 5
Clear The Depths through stage 5
Collect all Puzzle Pieces in Dream Fields
Collect all Puzzle Pieces in Bamboo Forest of the Lost
Collect all Puzzle Pieces in Azure Winter
Collect all Puzzle Pieces in Forest of Magic
Collect all Puzzle Pieces in Ultramarine Rain
Collect all Puzzle Pieces in Outside World
Collect all Puzzle Pieces in Scarlet Devil Mansion Interior
Collect all Puzzle Pieces in Shining Needle Castle
Collect all Puzzle Pieces in Dream Scraps
Collect all Puzzle Pieces in The Depths
Clear your first stage
Collect your first Puzzle Piece
Earn your first C Rank
Earn your first B Rank
Earn your first A Rank
Earn your first S Rank
See the ending
See the true ending
Summon over 100 heads
Retry over 100 times
Die over 100 times
Clear stages over 100 times in total
Clear a stage on Ninth Head Mode
Earn an S Rank in Jelly Attack!
Find the hidden stage
Complete the Banki Ice Cream
Clear stages 1-1 through 10-5 on Normal Mode
Complete any stage on Normal Mode with 1 extra head (except 1-1)
Have a Total Time of less than 30 minutes
Earn all achievements