A variety of interesting structures and dynamics stem from singularities of Poisson operators. As well known, the kernel of a finite-dimensional Poisson operator can be integrated “locally” to define Casimirs, and their level-sets foliate the phase space yielding symplectic submanifolds (Lie-Darboux theorem). However, the global structure can be more complex when the Poisson operator has “singularities” where its rank changes. Invoking some physical examples (both finite dimensional and infinite dimensional), we discuss how singularities cause seemingly non-Hamiltonian phenomena. Some theoretical methods for probing into singularities will be also introduced.