Here is the full list of all 27 Lemmings Touch trophies - there are 12 bronze, 8 silver, 6 gold and 1 platinum.
Obtain All Trophies
Complete All Hard Levels
Obtain All the stars from the Hard Levels
Complete All Taxing Levels
Obtain all the stars from the Taxing Levels
Obtain 290 Stars or more
Complete All Tricky Levels
Obtain all the stars from the Tricky Levels
Complete 50 Objectives
Unlock Hard Difficulty
Buy 20 items from the Look & Style menu
Complete All Easy Levels
Obtain all the stars from the Easy Levels
Unlock Taxing Difficulty
Buy 10 items from the Look & Style menu
Business Costume: Black Body/Hair/Moustache/Umbrella, Bowler Hat and Tie.
Complete 25 Objectives
Play 20 levels with any item from the Look & Style menu
Wizard Costume: Black Hair/Body, Wizard Hat, Glasses and Wand.
Unlock Tricky Difficulty
Complete 10 Objectives
Play a level with any item from the Look & Style menu
Buy your first item from the Look & Style menu
Complete the Tutorials
Kill your first Mischievous Lemming
Complete 1 Objective
Obtain All Stars on a level