  We prove that a holonomic binomial $D$--module $M_A (I,\beta)$ is regular if
and only if certain associated primes of $I$ determined by the parameter vector
$\beta\in \CC^d$ are homogeneous. We further describe the slopes of
$M_A(I,\beta)$ along a coordinate subspace in terms of the known slopes of some
related hypergeometric $D$--modules that also depend on $\beta$. When the
parameter $\beta$ is generic, we also compute the dimension of the generic
stalk of the irregularity of $M_A(I,\beta)$ along a coordinate hyperplane and
provide some remarks about the construction of its Gevrey solutions.
