Given a free resolution of an ideal $\mathfrak{a}$ of holomorphic functions, there is an associated residue current $R$ that coincides with the classical Coleff-Herrera product if $\mathfrak{a}$ is a complete intersection ideal and whose annihilator ideal equals $\mathfrak{a}$. In the case when $\mathfrak{a}$ is an Artinian monomial ideal, we show that the singularities of R are small in a certain sense if and only if $\mathfrak{a}$ is integrally closed.