Origami crease patterns have inspired the design of reconfigurable materials that can rigidly fold and provide structural stiffness in certain directions. Their deployability, however, can occur only in one direction along which they lose load-bearing capacity. Here, we introduce a multimodal class of rigid-foldable materials that can deploy into multiple directions and lock up into several stiff states due to panel self-contact. Hallmarks include omnidirectional structural resistance under compression, access to multiple flat-foldable and lockable folding paths, and capacity to undergo topological changes each with its own set of structural properties. Their high versatility can be used for structural applications across the length-scale spectrum, such as in-situ deployment of load-bearing structures, reconfigurable packaging, and miniaturized mechanical memories.