Summary: We study a singular parabolic equation of the total variation type in one dimension. The problem is a simplification of the singular curvature flow. We show the existence and uniqueness of weak solutions. We also prove the existence of weak solutions to the semi-discretization of the problem as well as convergence of the approximating sequences. The semi-discretization shows that facets must form. For a class of initial data we are able to study in detail the facet formation and interactions and their asymptotic behaviour. We note that our qualitative results may be interpreted with the help of a special composition of multivalued operators.