  We construct a method to obtain the algebraic classification of Poisson
algebras defined on a commutative associative algebra, and we apply it to
obtain the classification of the $3$-dimensional Poisson algebras. In addition,
we study the geometric classification, the graph of degenerations and the
closures of the orbits of the variety of $3$-dimensional Poisson algebras.
Finally, we also study the algebraic classification of the Poisson algebras
defined on a commutative associative null-filiform or filiform algebra and, to
enrich this classification, we study the degenerations between these particular
Poisson algebras.
