We consider the possibility that the Planck constant is characterized by stochastic fluctuations. The scenario is motivated by the Dirac idea that fundamental constant are dynamical variables and by conjectures on quan- tum structure of spacetime at small distances. We assume that the has time dependence consisting in gaussian random fluctuations around its con- stant mean value, with a typical correlation time scale τ. We consider the case of propagation of a free particle and a one–dimensional harmonic oscilla- tor coherent state, and find that the time evolution in both cases is different from the standard behaviour. Finally, we discuss how experiments exploiting coherent electromagnetic fields in a cavity may put effective bounds on the value of τ, and discuss other possible experimental signatures in interferometric experiments.