  Let $X$ be a Hirzebruch surface, and let $H$ be any ample divisor. In this
paper, we algorithmically determine when the moduli space of semistable sheaves
$M_{X,H}(r,c_1,c_2)$ is nonempty. Our algorithm relies on certain stacks of
prioritary sheaves. We first solve the existence problem for these stacks and
then algorithmically determine the Harder-Narasimhan filtration of the general
sheaf in the stack. In particular, semistable sheaves exist if and only if the
Harder-Narasimhan filtration has length one.
  We then study sharp Bogomolov inequalities $\Delta \geq \delta_H(c_1/r)$ for
the discriminants of stable sheaves which take the polarization and slope into
account; these inequalities essentially completely describe the characters of
stable sheaves. The function $\delta_H(c_1/r)$ can be computed to arbitrary
precision by a limiting procedure. In the case of an anticanonically polarized
del Pezzo surface, exceptional bundles are always stable and $\delta_H(c_1/r)$
is computed by exceptional bundles. More generally, we show that for an
arbitrary polarization there are further necessary conditions for the existence
of stable sheaves beyond those provided by stable exceptional bundles. We
compute $\delta_H(c_1/r)$ exactly in some of these cases. Finally, solutions to
the existence problem have immediate applications to the birational geometry of
moduli spaces of sheaves.
