  We study singular hermitian metrics on holomorphic vector bundles, following
Berndtsson-P{\u{a}}un. Previous work by Raufi has shown that for such metrics,
it is in general not possible to define the curvature as a current with measure
coefficients. In this paper we show that despite this, under appropriate
codimension restrictions on the singular set of the metric, it is still
possible to define Chern forms as closed currents of order 0 with locally
finite mass, which represent the Chern classes of the vector bundle.
