  In this article we study Cohen-Macaulay modules over one-dimensional
hypersurface singularities and the relationship with the representation theory
of associative algebras using methods of cluster tilting theory. We give a
criterion for existence of cluster tilting objects and their complete
description by homological methods, using higher almost split sequences and
results from birational geometry. We obtain a large class of 2-CY tilted
algebras which are finite dimensional symmetric and satisfy $\tau^2=\id$. In
particular, we compute 2-CY tilted algebras for simple and minimally elliptic
curve singularities.
