Where do butterflies sleep?
It would be better to say that they go into an inactive, low metabolic state. This low metabolic state is often driven by the temperature in the air itself; ectothermic butterflies require outside heat-energy to become active. Basically they use this time to digest their food and produce sperm/eggs. (reference). At night, or during inclement weather, most butterflies perch on the underside of a leaf (this provides them with a bit of shelter from the rain and makes them less noticeable to prey), crawl deep between blades of grass or into a crevice in rocks, or find some other shelter to rest (reference).  In short, I would not call it sleeping but just a state of rest. Butterflies cant even close their eyes from the lack of eyelids (reference).