  Let (M,g) be a compact oriented Riemannian manifold with an incomplete edge
singularity. This article shows that it is possible to evolve g by the Yamabe
flow within a class of singular edge metrics. As the main analytic step we
establish parabolic Schauder-type estimates for the heat operator on certain
H\"older spaces adapted to the singular edge geometry. We apply these estimates
to obtain local existence for a variety of quasilinear equations, including the
Yamabe flow. This provides a setup for a subsequent discussion of the Yamabe
problem using flow techniques in the singular setting.
