Here is the full list of all 13 The Sokoban trophies - there are 9 bronze, 3 silver and 1 gold.
Cleared a stage for the first time.
Used an answer ticket for the first time.
Solved a puzzle with the official perfect answer.
Unlocked puzzle set number 5.
Unlocked puzzle set number 8.
Unlocked puzzle set number 11.
Unlocked puzzle set number 14.
Unlocked puzzle set number 17.
Unlocked puzzle set number 20.
Cleared all stages of the hidden puzzle set 21.
Cleared all stages of the hidden puzzle set 22.
Cleared all stages of the hidden puzzle set 23.
Cleared all the puzzle sets.