Super-swords are multipurpose tools used to chop, well, everything, from grass to giant trees and stones. They are used automatically if you don’t have enough tools to cut a certain object.
You can buy Super-swords with Zombucks, get them as a prize for completing a quest, or win in one of the mini-games on the official Zombie Castaways Facebook page http://fb.com/ZombieCastaways/