A typical copula is singular

We present Baire category results in the class of bivariate copulas (or, equivalently, doubly stochastic probability measures) endowed with two different metrics under which the space is complete. Main content of the paper is that, in the sense of Baire categories with respect to the topology induced by the uniform metric, the family of absolutely continuous copulas is of first category, whereas the family of purely singular copulas is co-meager and, hence, of second category. Moreover, several other popular dense sub-classes of copulas are considered, like shuffles of Min and checkerboard copulas.