  In this article we classify quadruple Galois canonical covers $\phi$ of
singular surfaces of minimal degree. This complements the work done in
math.AG/0302045, so the main output of both papers is the complete
classification of quadruple Galois canonical covers of surfaces of minimal
degree, both singular and smooth. Our results show that the covers $X$ studied
in this article are all regular surfaces and form a bounded family in terms of
geometric genus $p_g$. In fact, the geometric genus of $X$ is bounded by 4.
Together with the results of Horikawa and Konno for double and triple covers, a
striking numerology emerges that motivates some general questions on the
existence of higher degree canonical covers. In this article, we also answer
some of these questions. The arguments to prove our results include a delicate
analysis of the discrepancies of divisors in connection with the ramification
and inertia groups of $\phi$.
