  We find a complete set of local invariants of singular symplectic forms with
the structurally stable Martinet hypersurface on a $2n$-dimensional manifold.
In the $\mathbb C$-analytic category this set consists of the Martinet
hypersurface $\Sigma _2$, the restriction of the singular symplectic form
$\omega$ to $T\Sigma_2$ and the kernel of $\omega^{n-1}$ at the point $p\in
\Sigma_2$. In the $\mathbb R$-analytic and smooth categories this set contains
one more invariant: the canonical orientation of $\Sigma_2$. We find the
conditions to determine the kernel of $\omega^{n-1}$ at $p$ by the other
invariants. In dimension $4$ we find sufficient conditions to determine the
equivalence class of a singular symplectic form-germ with the structurally
smooth Martinet hypersurface by the Martinet hypersurface and the restriction
of the singular symplectic form to it. We also study the singular symplectic
forms with singular Martinet hypersurfaces. We prove that the equivalence class
of such singular symplectic form-germ is determined by the Martinet
hypersurface, the canonical orientation of its regular part and the restriction
of the singular symplectic form to its regular part if the Martinet
hypersurface is a quasi-homogeneous hypersurface with an isolated singularity.
