We present both old and recent results about singular copulas and copulas with a singular component by discussing their relevance in (at least) three different domains. First, singular copulas may be used to obtain specific tail behavior in a multivariate distribution, a fact that has also been exploited to obtain worst-possible scenarios for risk measures. Second, special classes of singular copulas (e.g. shuffles of Min) can be used in the approximation of various dependence structures. Third, several copulas of previous type are solutions of optimization problems involving distributions with fixed marginals.