We consider the problem to minimize n-dimensional area among currents T whose boundary (or part of it) is supposed to lie in a given hypersurface S of \({\mathbb{R}}^{n+1}\). It is shown that the set of singular points of T - i.e. the closure of those points where T is not an embedded submanifold with boundary - has Hausdorff-dimension at most n-7. As in the case of interior regularity this result turns out to be optimal. The methods used are those of Geometric Measure Theory. The proof uses reflection arguments combined with regularity results (by Grüter and Jost) for varifolds with free boundaries. The existence of singular points is reduced to the existence of singular minimizing cones by Federer's dimension reduction argument.