  We use Numerical Linked Cluster Expansions (NLCE) to study the site diluted
transverse-field Ising model on the square-lattice at $T=0$. NLCE with a
self-consistent mean-field on the boundary of the clusters is used to obtain
the ground state magnetization, susceptibility, and structure factor as a
function of transverse field $h$ and exchange constant $J$. Adding
site-dilution to the model turns NLCE into a series expansion in the dilution
parameter $p$. Studying the divergence of the structure factor allows us to
establish the phase-diagram in the $h/J$ and $p$ plane. By studying the
magnetization of the system in a longitudinal field, we investigate the
Griffiths-McCoy (GM) singularities. We find that the magnetization develops
non-linearities in the Griffiths phase with exponents that vary continuously
with $h$. Additionally, the probability distribution of the local
susceptibility develops long tails in the Griffith's phase, which is studied in
terms of its moments.
