  Isolated hypersurface singularities come equipped with a Milnor lattice, a
${\mathbb Z}$-lattice of finite rank, and a set of $distinguished$ ${\mathbb
Z}$-bases of this lattice. Usually these bases are constructed from $one$
morsification and $all\ possible$ choices of distinguished systems of paths.
But what does one obtain if one considers $all\ possible$ morsifications and
$one$ fixed distinguished system of paths? Looijenga asked this question 1974
for the simple singularities. He and Deligne found that one obtains a bijection
between Stokes regions in a universal unfolding and the set of distinguished
bases modulo signs. This allows to see the base space of the universal
unfolding as an atlas of Stokes data. Here we reprove their result and extend
it to the simple elliptic singularities. We use more conceptual arguments,
moduli spaces of marked singularities (i.e. Teichm\"uller spaces for
singularities), extensions of them to F-manifolds, and the actions of
symmetries of singularities on the Milnor lattices and these moduli spaces. We
use and extend results of Jaworski on the Lyashko-Looijenga maps for the simple
elliptic singularities. The sections 2 and 3 give a survey on singularities and
the associated objects which allows to read the paper independently of other
sources.
