A number of exciting problems in solid mechanics are formulated as unstable variational problems. Among them: structural optimization, bounds on composite properties, phase transitions in solids, inverse problems of determination of material's structure and of nondestructive testing. Solutions of these problems are characterized by fine scale spatial inhomogenuities that come from the non-quasiconvexity of Lagrangians and that are realized as media with microstructures. Dealing with these problems, one has to determine "the best" structure of a material. We discuss methods for analysis of unstable problems, especially the technique of necessary conditions, and the translation method for sufficient condition. These methods establish averaged constituence relations in an optimal structure, and provide conditions for the fields in each material inside the mixture. The applications deal with optimal micro-geometries of multicomponent mixtures. Also, we discuss dynamics of a transition in natural unstable systems that leads to a micro-inhomogeneous equilibrium. This dynamics is characterized by oscillations that transform energy to a high-frequency mode, which leads to energy dissipation. The modelling and homogenization of a discrete chain of masses and unstable springs is discussed.