  We study versions of homological mirror symmetry for hypersurface cusp
singularities and the three hypersurface simple elliptic singularities. We show
that the Milnor fibres of each of these carries a distinguished Lefschetz
fibration; its derived directed Fukaya category is equivalent to the derived
category of coherent sheaves on a smooth rational surface $Y_{p,q,r}$. By using
localization techniques on both sides, we get an isomorphism between the
derived wrapped Fukaya category of the Milnor fibre and the derived category of
coherent sheaves on a quasi-projective surface given by deleting an
anti-canonical divisor $D$ from $Y_{p,q,r}$. In the cusp case, the pair
$(Y_{p,q,r}, D)$ is naturally associated to the dual cusp singularity, tying
into Gross, Hacking and Keel's proof of Looijenga's conjecture.
