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
Obtain 290 Stars or more
Complete 50 Objectives
Buy 20 items from the Look & Style menu
Obtain all the stars from the Taxing Levels
Complete All Taxing Levels
Complete 25 Objectives
Wizard Costume: Black Hair/Body, Wizard Hat, Glasses and Wand.
Business Costume: Black Body/Hair/Moustache/Umbrella, Bowler Hat and Tie.
Buy 10 items from the Look & Style menu
Complete All Tricky Levels
Obtain all the stars from the Tricky Levels
Play 20 levels with any item from the Look & Style menu
Complete 10 Objectives
Play a level with any item from the Look & Style menu
Buy your first item from the Look & Style menu
Unlock Hard Difficulty
Obtain all the stars from the Easy Levels
Complete All Easy Levels
Unlock Taxing Difficulty
Unlock Tricky Difficulty
Complete the Tutorials
Kill your first Mischievous Lemming
Complete 1 Objective
Obtain All Stars on a level