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Instabilities can lead to surprising and complex behaviors in soft matter and biological systems. In this thesis, we study this theme in two contexts: ecological competition in a turbulent ocean, and patterns of localized swelling and shrinking in thin elastic sheets. In the first half of the thesis, we explore evolutionary dynamics in the presence of turbulent advection, inspired by the behavior of phytoplankton at oceanic fronts. We focus on the impact of an effective compressibility due to vertical velocity gradients in the flow acting on communities living at a fixed depth. We first consider fixation probabilities in idealized one-dimensional flows, and then study changes in the relative abundances of populations in realistic flow fields constructed using observational data. In both cases, the flow makes the fate of organisms strongly dependent on local flow conditions, with organisms born in upwelling regions experiencing a substantial advantage. In the second half of the thesis, we investigate membranes with arrays of impurities that create localized regions of expansion or contraction. When the size mismatch is sufficiently large or the sheet is sufficiently flexible, the regions of dilation buckle into the third dimension, choosing their buckling direction based on interactions with their neighbors. The buckled nodes behave like discrete, but highly compressible, Ising spins with antiferromagnetic interactions. Upon including thermal fluctuations, we observe a phase transition accompanied by an unusual divergence and sign change in the coefficient of thermal expansion.