  We consider a simple model for the growth of isolated steps on a vicinal
crystal surface. It incorporates diffusion and drift of adatoms on the terrace,
and strong step and kink edge barriers. Using a combination of analytic methods
and Monte Carlo simulations, we study the morphology of growing steps in
detail. In particular, under typical Molecular Beam Epitaxy conditions the step
morphology is linearly unstable in the model and develops fingers separated by
deep cracks. The vertical roughness of the step grows linearly in time, while
horizontally the fingers coarsen proportional to $t^{0.33}$. We develop scaling
arguments to study the saturation of the ledge morphology for a finite width
and length of the terrace.
