  Poincar\'e recurrence theorem implies the density of recurrent points for
volume-preserving dynamical systems on compact domains. The density of closed
orbits in the non-wandering set is one of the essential properties of Axiom A
and chaos. The minimal flows are one of the most fundamental objects in
topological dynamics. To analyze minimality, recurrence, and density in this
paper, we introduce a strict limit circuit and a circuit with wandering
holonomy. Using these concepts, we classify non-recurrent non-wandering orbits
of flows on compact surfaces with finitely many singular points, characterize
the minimality of flows on compact surfaces, and generalize the Poincar\'e
recurrence theorem for flows on surfaces. Moreover, we characterize the density
of closed orbits in the non-wandering set under finiteness of singular points.
Furthermore, we demonstrate the necessity of finiteness of singular points and
compactness of surfaces. In addition, the analogous results hold for flows with
finitely many connected components of the singular point set on non-compact
surfaces using the end compactification and collapsing singular points.
