Here is the full list of all 18 Yoko & Yuki: Dr. Rat's Revenge trophies - there are 3 bronze, 5 silver, 9 gold and 1 platinum.
Unlock all trophies
Complete level one
Complete world one
Complete world two
Complete world three
Obtain the gravitron
Obtain scissors
Obtain the paint
Obtain the tangle
Turn the world for 72 times
Turn the world for 200 times
Turn the world for 489 times
Die for 30 times
Die for 60 times
Die for 111 times
Jump for 150 times
Jump for 400 times
Stay idle for three minutes