  We prove an Allard-type regularity theorem for free-boundary minimal surfaces
in Lipschitz domains locally modelled on convex polyhedra. We show that if such
a minimal surface is sufficiently close to an appropriate free-boundary plane,
then the surface is $C^{1,\alpha}$ graphical over this plane. We apply our
theorem to prove partial regularity results for free-boundary minimizing
hypersurfaces, and relative isoperimetric regions.
