Finding all circuits that contain a given edge
It should be NP-complete to compute, given an arc $e = uv$ of a directed graph $D = (V,A)$ whether there is a cycle containing some vertex $w$ using the arc $e$.  The instance to this problem is $(D,e,w)$.

By reducing from back-and-forth (two-disjoint paths from $a$ to $b$ and from $b$ to $a$), given an instance $(D',a,b)$ of back-and-forth, construct the instance $D$ where you make two copies of $a$, $a_1$ and $a_2$ and an arc going from $a_1$ to $a_2$.  The constructed instance is $(D,(a_1a_2),b)$.

Now, suppose there are two-disjoint paths from $a$ to $b$ and back, then there is a cycle from $a_2$ to $b$ and from $b$ to $a_1$, hence $b$ is on a cycle containing the arc $a_1a_2$.  For the reverse direction suppose that $b$ is on a cycle traversing the arc $a_1a_2$.  Then, in the original graph, there is a path from $a$ to $b$ and one from $b$ to $a$.  qed