The systematic numerical studies of hydrodynamic instability of axisymmetrical liquid sheets interacting with surroundings across internal and external surfaces have been carried out. The algorithms for calculation of the amplification factors and other parameters of unstable waves have been created in the frame of Euler's equations and Navier-Stokes equations for expanding sheets and compound capillary jets. The problem formulation takes into account the translation movement in liquid layers and its surroundings, the spinning of all mediums, the axisymmetrical and the angle periodic perturbations for arbitrary wave numbers. Densities and viscosities of all mediums and surface tensions take into consideration too. It has been found that the variation of governing parameters and flow spinning effects to the type of dominating perturbations.