  We define Poincar\'e series associated to a toric or analytically irreducible
quasi-ordinary hypersurface singularity, (S,0), by a finite sequence of
monomial valuations, such that at least one of them is centered at the origin
0. This involves the definition of a multi-graded ring associated to the
analytic algebra of the singularity by the sequence of valuations. We prove
that the Poincar\'e series is a rational function with integer coefficients,
which can be defined also as an integral with respect of the Euler
characteristic, over the projectivization of the analytic algebra of the
singularity, of a function defined by the valuations. In particular, the
Poincar\'e series associated to the set of divisorial valuations associated to
the essential divisors, considered both over the singular locus and over the
point 0, is an analytic invariant of the singularity. In the quasi-ordinary
hypersurface case we prove that this Poincar\'e series determines and it is
determined by the normalized sequence of characteristic monomials. These
monomials in the analytic case define a complete invariant of the embedded
topological type of the hypersurface singularity.
