  We study tautological vector bundles over the Hilbert scheme of points on
surfaces. For each K-trivial surface, we write down a simple criterion ensuring
that the tautological bundles are big and nef, and illustrate it by examples.
In the K3 case, we extend recent constructions and results of Bini, Boissi\`ere
and Flamini from the Hilbert scheme of 2 and 3 points to an arbitrary number of
points. Among the K-trivial surfaces, the case of Enriques surfaces is the most
involved. Our techniques apply to other smooth projective surfaces, including
blowups of K3s and minimal surfaces of general type, as well as to the punctual
Quot schemes of curves.
