  We study the distribution of resonances for discrete Hamiltonians of the form
$H_0+V$ near the thresholds of the spectrum of $H_0$. Here, the unperturbed
operator $H_0$ is a multichannel Laplace type operator on $\ell^2(\mathbb Z;
\mathfrak{G}) \cong \ell^2(\mathbb Z)\otimes \mathfrak{G}$ where ${\mathfrak
G}$ is an abstract separable Hilbert space, and $V$ is a suitable
non-selfadjoint compact perturbation. We distinguish two cases. If ${\mathfrak
G}$ is of finite dimension, we prove that resonances exist and do not
accumulate at the thresholds in the spectrum of $H_0$. Furthermore, we compute
exactly their number and give a precise description on their location in
clusters in the complex plane. If ${\mathfrak G}$ is of infinite dimension, an
accumulation phenomenon occurs at some thresholds. We describe it by means of
an asymptotical analysis of the counting function of resonances. Consequences
on the distribution of the complex and the embedded eigenvalues are also given.
