Let \(R=P/(f)\) be an analytic hypersurface ring. The author investigates first the relation between maximal Cohen-Macaulay modules (MCM) over R and over \(R_ 1=P_ 1/(f+y^ 2)\), where \(P_ 1=P\) (and \(P=k\), k algebraically closed and char(k)\(\neq 2)\). He proves in \(corollary^ 2.8\) that there are only finitely many isomorphism classes of indecomposable MCM's over R if and only if this is true for \(R_ 1\). In theorem 3.1 it is shown - in a more general frame - that there is a canonical bijection between the sets of isomorphism classes of MCM's over R and over \(R_ 2=P_ 2/(f+y^ 2+z^ 2)\) respectively, where \(P_ 2=P.\) Since the two-dimensional simple singularities, i.e. the rational double points, have only finitely many isomorphism classes of MCM's over their local rings [by \textit{M. Artin} and \textit{J.-L. Verdier}, Math. Ann. 270, 79-82 (1985; Zbl 0553.14001)], one gets by iterated application of corollary 2.8 the main result of this paper: There are only finitely many isomorphism classes of indecomposable MCM's over the local ring of an isolated simple hypersurface singularity \((A_ k, D_ k, E_ 6, E_ 7, E_ 8\) in Arnold's classification). The author also gives a conceptional description of the Auslander-Reiten quivers of the simple plane curve singularities in \(char(k)=0\), showing that these quivers coincide with certain graphs associated to representations of finite reflection groups in \(Gl(2,k).\) [See also the following review.]