We define an object called a G-struture, and use it to classify disconnected regular covering spaces of a base space in terms of group actions. We then apply this result to classify discrete group actions on singular surfaces whose quotient space is homeomorphic to an oriented closed surface. We also investigate the converse question: given an oriented closed surface with a set of singular points on it marked by stabilizer information, when does this result from a group action? For a large class of stabilizer data we have exact answers based the theory of orbifolds.