  In this paper we prove the existence of a simultaneous local normalization
for couples $(X,\mathcal{G})$, where $X$ is a vector field which vanishes at a
point and $\mathcal{G}$ is a singular underlying geometric structure which is
invariant with respect to $X$, in many different cases: singular volume forms,
singular symplectic and Poisson structures, and singular contact structures.
Similarly to Birkhoff normalization for Hamiltonian vector fields, our
normalization is also only formal, in general. However, when $\mathcal{G}$ and
$X$ are (real or complex) analytic and $X$ is analytically integrable or
Darboux-integrable then our simultaneous normalization is also analytic. Our
proofs are based on the toric approach to normalization of dynamical systems,
the toric conservation law, and the equivariant path method. We also consider
the case when $\mathcal{G}$ is singular but $X$ does not vanish at the origin.
