  This paper is part of a series of papers on differential geometry of
$C^\infty$-ringed spaces. In this paper, we study vector fields and their flows
on a class of singular spaces. Our class includes arbitrary subspaces of
manifolds, as well as symplectic and contact quotients by actions of compact
Lie groups. We show that derivations of the $C^\infty$-ring of global smooth
functions integrate to smooth flows.
