  This paper includes results centered around three topics, all of them related
with the nonlinear stability of equilibria in Poisson dynamical systems.
Firstly, we prove an energy-Casimir type sufficient condition for stability
that uses functions that are not necessarily conserved by the flow and that
takes into account certain asymptotically stable behavior that may occur in the
Poisson category. This method is adapted to Poisson systems obtained via a
reduction procedure and we show in examples that the kind of stability that we
propose is appropriate when dealing with the stability of the equilibria of
some constrained systems. Finally, we discuss two situations in which the use
of continuous Casimir functions in stability studies is equivalent to the
topological stability methods introduced by Patrick {\it et al.}
