  In two parts, we present a bigness criterion for the cotangent bundle of
resolutions of orbifold surfaces of general type. As a corollary, we obtain the
\textit{canonical model singularities} (CMS) criterion that can be applied to
determine when a birational class of surfaces has smooth representatives with
big cotangent bundle and compare it with other known criteria. We then apply
this criterion to the problem of finding the minimal degrees $d$ for which the
deformation equivalence class of a smooth hypersurface of degree d in $\PP^3$
has a representative with big cotangent bundle; applications to the minimal
resolutions of cyclic covers of $\PP^2$ branched along line arrangements in
general position are also obtained.
  The CMS criterion involves invariants of canonical singularities whose values
were unknown. In this part of the work, we describe a method of finding these
invariants and obtain formulas for $A_n$-singularities. We also use our
approach to derive several extension results for symmetric differentials on the
complement of the exceptional locus $E$ of the minimal resolution of an
$A_n$-singularity; namely, we characterize the precise extent to which the
poles along $E$ of symmetric differentials on the complement are milder than
logarithmic poles.
