  The trace of the canonical module of a Cohen-Macaulay ring describes its
non-Gorenstein locus. We study the trace of the canonical module of a Segre
product of algebras, and we apply our results to compute the non-Gorenstein
locus of toric rings. We provide several sufficient and necessary conditions
for Hibi rings and normal semigroup rings to be Gorenstein on the punctured
spectrum.
