Here is the full list of all 12 Lemmings (JP) trophies - there are 8 bronze, 3 silver and 1 gold.
Send 1000 Lemmings to their death
Blow up 100 Lemmings
Save every Lemming possible
Complete all Mayhem levels
Make a clone that can float and climb
Complete all Taxing levels
Set off 100 digger Lemmings
Beat 3rd Time Lucky in under 1minute 30 seconds
Complete all Tricky levels
Build 100 bridges
Complete all Fun levels
Complete Teach me Climbers and Floaters in under one minute