Abstract: This thesis provides upper bounds for a certain singularity invariant, in the case where the underlying ring is in one of a few well-known classes of rings. The Hartshorne-Speiser-Lyubeznik number or HSL number of a Noetherian local (or graded) ring is a measure of the nilpotency of Frobenius on local cohomology with support at the (homogeneous) maximal ideal. It is known that the HSL number of such a ring is finite, but it is not known in general how to compute the HSL number for any given ring. In this thesis we present computable upper bounds for the HSL numbers of semigroup rings, toric face rings, and quotients of polynomial rings by monomial ideals, and compute the exact HSL number of monomial hypersurfaces. We also relate the HSL number of a ring to the HSL number of its reduction, provide Macaulay2 code to compute HSL numbers of quotients of polynomial rings, and discuss the application of our results to bounding Frobenius test exponents.