  In this paper, we focus on the initial degree and the vanishing of the
Valabrega-Valla module of a pair of monomials ideals $J\subseteq I$ in a
polynomials ring over a field $\mathbb{K}$. We prove that the initial degree of
this module is bounded above by the maximum degree of a minimal generators of
$J$. For edge ideals of graphs, a complete characterization of the vanishing of
the Valabrega-Valla module is given. For higher degree ideals, we find classes
which the Valabrega-Valla module vanishes. For the case that $J$ is the facet
ideal of a clutter $\mathcal{C}$ and $I$ is the defining ideal of singular
subscheme of $J$, the non-vanishing of this module is investigated in terms of
the combinatorics of $\mathcal{C}$. Finally, we describe the defining ideal of
the Rees algebra of $I/J$ provided that the Valabrega-Valla module is zero.
