We consider $n$-dimensional hypersurfaces flowing by mean curvature flow with
Neumann free boundary conditions supported on a smooth support surface. We show
that the Hausdorff $n$-measure of the singular set is zero. In fact, we
consider two types of interaction between the support and flowing surfaces. In
the case of weaker interaction, we need make no further assumptions than in the
case without boundary to achieve our result. In the case of stronger
interaction, we need only make the additional assumption that $H_{\Sigma}>0$,
that is, that the support surface be mean convex.