  The Hilbert series of a polarised algebraic variety $(X,D)$ is a powerful
invariant that, while it captures some features of the geometry of $(X,D)$
precisely, often cannot recover much information about its singular locus. This
work explores the extent to which the Hilbert series of an orbifold del Pezzo
surface fails to pin down its singular locus, which provides nonexistence
results describing when there are no orbifold del Pezzo surfaces with a given
Hilbert series, supplies bounds on the number of singularities on such
surfaces, and has applications to the combinatorics of lattice polytopes in the
toric case.
