  We study virtual invariants of Quot schemes parametrizing quotients of
dimension at most 1 of the trivial sheaf of rank N on nonsingular projective
surfaces. We conjecture that the generating series of virtual K-theoretic
invariants are given by rational functions. We prove rationality for several
geometries including punctual quotients for all smooth projective surfaces and
dimension 1 quotients for surfaces X with p_g>0. We also show that the
generating series of virtual cobordism classes can be irrational.
  Given a K-theory class on X of rank r, we associate natural series of virtual
Segre and Verlinde numbers. We show that the Segre and Verlinde series match in
the following three cases: Quot schemes of dimension 0 quotients, Hilbert
schemes of points and curves over surfaces with p_g>0, Quot schemes of minimal
elliptic surfaces for quotients supported on fiber classes. Moreover, for
punctual quotients of the trivial sheaf of rank N, we prove a new symmetry of
the Segre/Verlinde series exchanging r and N. The Segre/Verlinde statements
have analogues for punctual Quot schemes over curves.
