The present work addresses static, quasi-static and implicit transient structural non-linear stability optimization having the relative density representing the topology, elemental thicknesses for sizing of shell parts or the nodal positions of the finite elements as design variables. The discrete adjoint sensitivities are implemented for addressing industrial applications using as a general non-linear FE-solver. The FE-modeling includes non-linear constitutive materials, geometrical non-linear deformations and general contacts between parts. The forward pass solving the primal solutions applies backward Euler and Newmark time integration for the quasi-static and the transient structural modeling, respectively. Thereby, the global operators of the forward pass for the primal solution are reused for the discrete adjoint backward integration. Common optimization approaches are to simplify the physics and ignore the non-linear effects of the realistic physical modelling capturing the structural non-linear stability. However, such simplifications often fails when the linear optimized structures are validated using full non-linear modelling. Hence, the present work shows for a number of both simple academic benchmarks and industrial applications that sensitivity-based optimization approaches using various recent stabilization techniques can successfully address non-parametric optimizations where severe non-linear stability issues are present in the structural modelling both with respect to the numerical stability and for capturing the correct physical structural stability in a given optimization formulation. Furthermore, the non-parametric optimization results will demonstrate the importance of applying sufficient geometrical imperfections when applying large deformation modeling for non-linear structural stability optimization.