Let $S$ be a generic cubic surface and let $C$ be its intersection with a generic quadric surface.
In the linear system of hyperplane sections of $S$, how many points represent the planes $H$ tangent to $C$ at two points, such that the curve $H \cap S$ is singular?



Points of a linear system on a cubic surface
OK, I'll try to answer your corrected question : you are looking at planes which are bitangent to $C$, and tangent to $S$ somewhere. Now these are two independent properties, so in the dual $(\mathbb{P}^3)^\*$ your planes are the intersection points of the dual surface $S^\*$ and the curve of bitangent planes, say $\Gamma $.


The degree of $S^\*$ is $3.2^2=12$. To compute the degree of $\Gamma $ one counts its intersection with a general plane in $(\mathbb{P}^3)^\*$, that is, bitangent planes to $C$ passing through a general point $p$ of $\mathbb{P}^3$. From $p$ $C$ projects to a plane sextic $C'$ with 6 nodes, and the bitangent planes through $p$ to bitangent lines to $C'$. Their number is given by the Plücker formula; if I am not mistaken it is 12. So the answer is $12^2=144$. 


