  We study Ruan's \textit{cohomological crepant resolution conjecture} for
orbifolds with transversal ADE singularities. In the $A_n$-case we compute both
the Chen-Ruan cohomology ring $H^*_{\rm CR}([Y])$ and the quantum corrected
cohomology ring $H^*(Z)(q_1,...,q_n)$. The former is achieved in general, the
later up to some additional, technical assumptions. We construct an explicit
isomorphism between $H^*_{\rm CR}([Y])$ and $H^*(Z)(-1)$ in the $A_1$-case,
verifying Ruan's conjecture. In the $A_n$-case, the family
$H^*(Z)(q_1,...,q_n)$ is not defined for $q_1=...=q_n=-1$. This implies that
the conjecture should be slightly modified. We propose a new conjecture in the
$A_n$-case which we prove in the $A_2$-case by constructing an explicit
isomorphism.
