  In this article, we consider weak del Pezzo surfaces defined over a finite
field, and their associated, singular, anticanonical models. We first define
arithmetic types for such surfaces, by considering the Frobenius actions on
their Picard groups; this extends the classification of Swinnerton-Dyer and
Manin for ordinary del Pezzo surfaces. We also show that some invariants of the
surfaces only depend on the above type.Then we study an inverse Galois problem
for singular del Pezzo surfaces having degree $3\leq d\leq 6$: we describe
which types can occur over a given finite field (of odd characteristic when
$3\leq d\leq 4$).
