The architectural features of branching morphogenesis demonstrate exquisite reproducibility among various organs and species despite the unique functionality and biochemical differences of their microenvironment. The regulatory networks that drive branching morphogenesis employ cell-generated and passive mechanical forces, which integrate extracellular signals from the microenvironment into morphogenetic movements. Cell-generated forces function locally to remodel the extracellular matrix (ECM) and control interactions among neighboring cells. Passive mechanical forces are the product of mechanical instabilities that trigger out-of-plane buckling and clefting deformations of adjacent tissues. Many of the molecular and physical signals that underlie buckling and clefting morphogenesis remain unclear and require new experimental strategies to be uncovered. Here, we highlight soft material systems that have been engineered to display programmable buckles and creases. Using synthetic materials to model physicochemical and spatiotemporal features of buckling and clefting morphogenesis might facilitate our understanding of the physical mechanisms that drive branching morphogenesis across different organs and species.