  We prove two partial regularity results for the scalar equation
$u_t+u_{xxxx}+\partial_{xx}u_x^2=0$, a model of surface growth arising from the
physical process of molecular epitaxy. We show that the set of space-time
singularities has (upper) box-counting dimension no larger than $7/6$ and
$1$-dimensional (parabolic) Hausdorff measure zero. These parallel the results
available for the three-dimensional Navier--Stokes equations. In fact the
mathematical theory of the surface growth model is known to share a number of
striking similarities with the Navier--Stokes equations, and the partial
regularity results are the next step towards understanding this remarkable
similarity. As far as we know the surface growth model is the only
lower-dimensional "mini-model" of the Navier--Stokes equations for which such
an analogue of the partial regularity theory has been proved. In the course of
our proof, which is inspired by the rescaling analysis of Lin (1998) and
Ladyzhenskaya & Seregin (1999), we develop certain nonlinear parabolic
Poincar\'e inequality, which is a concept of independent interest. We believe
that similar inequalities could be applicable in other parabolic equations.
