The objective of this paper is twofold. First, the problem of generation of real random matrix samples with uniform distribution in structured (spectral) norm bounded sets is studied. This includes an analysis of the distribution of the singular values of uniformly distributed real matrices, and an efficient (i.e. polynomial-time) algorithm for their generation. Second, it is shown how the developed techniques may be used to solve in a probabilistic setting several hard problems involving systems subject to real structured uncertainty.