  It is known that $C^r$ Morse-Smale vector fields form an open dense subset in
the space of vector fields on orientable closed surfaces and are structurally
stable for any $r \in \mathbb{Z}_{>0}$. In particular, $C^r$ Morse vector
fields (i.e. Morse-Smale vector fields without limit cycles) form an open dense
subset in the space of $C^r$ gradient vector fields on orientable closed
surfaces and are structurally stable. Therefore generic time evaluations of
gradient flows on orientable closed surfaces (e.g. solutions of differential
equations) are described by alternating sequences of Morse flows and
instantaneous non-Morse gradient flows. To illustrate the generic transitions,
we characterize and list all generic non-Morse gradient flows. To construct
such characterizations, we characterize isolated singular points of gradient
flows on surfaces. In fact, such a singular point is a non-trivial finitely
sectored singular point without elliptic sectors. Moreover, considering
Morse-Smale flows as "generic gradient flows with limit cycles", we
characterize and list all generic non-Morse-Smale flows.
