From Nookipedia, the Animal Crossing wiki
A Cicada Shell (セミのぬけがら, Semi no nuke gara) is a bug that can be caught in Animal Crossing: New Leaf. Technically, it is just the exoskeleton left behind by a cicada nymph, making it the only bug in the game that is not living. Cicada Shells can be found on trees; the player can run or walk past the trees without scaring the bug away. The only way to make the Cicada Shell disappear is to shake the tree or hit the tree with an axe, causing it to drop to the ground and disappear.
Donating to the Museum
In New Leaf
"Cicada larvae emerge from the ground in the evening and molt on trees, leaving behind empty shells. It's possible to find out the species of cicadas just by looking at these shells. Collecting the shells will tell you how many and what kind of cicadas inhabit a certain area."
It can be found in the top section of the upper right room in the bug habitat of the museum, with the other cicadas.
Names in other languages