In this dissertation, we study spatially localized pattern formations in highly deformable solids that can be characterized by their nonlinear elastic behavior. Three distinct problems are considered in this work: localized wrinkling of long beams resting on softening foundation, folding of beams on foundations and surface creasing in solids. In the context of localized wrinkling, we discuss a phenomenon associated with hidden symmetry that emerges asymptotically in finitely long beams on foundations, a problem having a finite symmetry group, when a certain non-dimensional length becomes very large. Later in this work, we study highly localized deformations that appear in the form of fold patterns in beams on elastic foundations and self-contacting surface creases on soft solids subjected to compression. We propose a robust computational methodology which is based on using bifurcation theory coupled with efficient numerical continuation techniques to compute such deformations. We make use of inherent symmetry in the system to work with appropriately chosen constraints and compute different symmetry-breaking bifurcation paths and study the stability of solutions on these paths. In particular, we show that folding and creasing are not local bifurcations from the state of homogeneous deformations of beams and solids respectively. They rather emerge as nonlinear deformations that evolve along global bifurcation paths.