  We consider symplectic singularities in the sense of A. Beauville as examples
of Poisson schemes. Using Poisson methods, we prove that a symplectic
singularity admits a finite stratification with smooth symplectic strata. We
also prove that in the formal neighborhood of a closed point in some stratum,
the singularity is a product of the stratum and a transversal slice. The
transversal slice is also a symplectic singularity, and the product
decomposition is compatible with natural Poisson structures. Moreover, we prove
that the transversal slice admits a $C^*$-action dilating the symplectic form.
