  Phase transition problems on curved surfaces can lead to a panopticon of
fascinating patterns. In this paper we consider finite element approximations
of phase field models with a spatially inhomogeneous and anisotropic surface
energy density. The problems are either posed in $\mathbb R^3$ or on a
two-dimensional hypersurface in $\mathbb R^3$. In the latter case, a
fundamental choice regarding the anisotropic energy density has to be made. Our
numerical method can be employed both situations, where for the problems on
hypersurfaces the algorithm uses parametric finite elements. We prove an
unconditional stability result for our schemes and present several numerical
experiments, including for the modelling of ice crystal growth on a sphere.
