  We construct several examples related to the scaling limits of energy
minimizers and gradient flows of surface energy functionals in heterogeneous
media. These include both sharp and diffuse interface models. The focus is on
two separate but related issues, the regularity of effective surface tensions
and the occurrence of zero mobility in the associated gradient flows. On
regularity we build on the theory of Goldman, Chambolle and Novaga to show that
gradient discontinuities in the surface tension are generic for sharp interface
models. In the diffuse interface case we only show that the laminations by
plane-like solutions satisfying the strong Birkhoff property generically are
not foliations and do have gaps. On mobility we construct examples in both the
sharp and diffuse interface case where the homogenization scaling limit of the
$L^2$ gradient flow is trivial, i.e. there is pinning at every direction. In
the sharp interface case, these are related to examples previously constructed
by Novaga and Valdinoci for forced mean curvature flow.
