We prove that every closed set which is not sigma-finite with respect to the
Hausdorff measure H^{N-1} carries singularities of continuous vector fields in
the Euclidean space R^N for the divergence operator. We also show that finite
measures which do not charge sets of sigma-finite Hausdorff measure H^{N-1} can
be written as an L^1 perturbation of the divergence of a continuous vector
field. The main tool is a property of approximation of measures in terms of the
Hausdorff content
