Equilibrium bifurcation is often employed as an effective mechanical strategy to design structures and materials capable to exhibit extreme and unconventional behaviours, from auxeticity (i.e. negative Poisson’s ratio) to tunable multi-stability. However, in classical structural instability problems, the intrinsic geometrical nonlinearity is balanced by hypotheses of linear elasticity for both continuous and discrete systems. Moreover, while compressive buckling has been known for a long time and has already found application in different fields, its tensile counterpart was only more recently discovered and its potential still needs to be fully revealed. Here, starting from some results obtained by the Authors, we analyse the role of nonlinear elasticity in overcoming possible limitations of linear constitutive assumptions as well as in conditioning the qualitative mechanical response and the critical loads of buckling structures, thus opening new perspectives for design and engineering application purposes. With this in mind, we show how the Feodosyev model, classically used as benchmark for describing the competition between axial deformation and compressive buckling, can be extended to study the effects of the interplay between hyperelasticity and large displacements on the bifurcation, then demonstrating how the same approach can also lead to define the first example of axially deformable system buckling under tension. Finally, with the aim of enriching the deformation class, we illustrate ad hoc conceived one- and bi-dimensional multi-scale paradigms incorporating discontinuities for highlighting how tensile buckling and prestress can be exploited in a way to tune the homogenized mechanical response, derived via structured deformations theory. It is felt that these findings might help to conceive new bifurcation-based architected materials and envisage alternative bio-inspired systems to be used in soft robotics applications as well as artificial prostheses for biomedical employment.