  We first give an alternative proof, based on a simple geometric argument, of
a result of Marian, Oprea and Pandharipande on top Segre classes of the
tautological bundles on Hilbert schemes of $K3$ surfaces equipped with a line
bundle. We then turn to the blow-up of $K3$ surface at one point and establish
vanishing results for the corresponding top Segre classes in a certain range.
This determines, at least theoretically, all top Segre classes of tautological
bundles for any pair $(\Sigma,H),\,H\in {\rm Pic}\,\Sigma$.
