On the Nash problem for terminal threefolds of type $cA\mathrm{/}r$

We study Nash valuations and essential valuations of terminal threefolds of type $cA\mathrm{/}r$. If $r=1$ or the given threefold is $\mathbb{Q}$-factorial, then all the Nash valuations and essential valuations can be completely described. We construct non-Gorenstein or non-$\mathbb{Q}$-factorial counter examples for the Nash problem.