Elucidating versatile configurations of spiral folding, and investigating the deployment performance is of relevant interest to extend the applicability of deployable membranes towards large-scale and functional configurations. In this paper we propose new schemes to package flat and curved membranes of finite thickness by using multiple spirals, whose governing equations render folding lines by juxtaposing spirals and by accommodating membrane thickness. Our experiments using a set of topologically distinct flat and curved membranes deployed by tensile forces applied in the radial and circumferential directions have shown that (1) the multi-spiral approach with prismatic folding lines offered the improved deployment performance, and (2) the deployment of curved surfaces progresses rapidly within a finite load domain. Furthermore, we confirmed the high efficiency of membranes folded by multi-spiral patterns. From viewpoints of configuration and deployment performance, the multi-spiral approach is potential to extend the versatility and maneuverability of spiral folding mechanisms.