In the mean field limit , isolated gravitational systems often evolve towards a steady state through a violent relaxation phase . One question is to understand the nature of this relaxation phase , in particular the role of radial instabilities in the establishment/destruction of the steady profile . Here , through a detailed phase-space analysis based both on a spherical Vlasov solver , a shell code and a N -body code , we revisit the evolution of collisionless self-gravitating spherical systems with initial power-law density profiles \rho ( r ) \propto r ^ { n } , 0 \leq n \leq - 1.5 , and Gaussian velocity dispersion . Two sub-classes of models are considered , with initial virial ratios \eta = 0.5 ( “ warm ” ) and \eta = 0.1 ( “ cool ” ) . Thanks to the numerical techniques used and the high resolution of the simulations , our numerical analyses are able , for the first time , to show the clear separation between two or three well known dynamical phases : ( i ) the establishment of a spherical quasi-steady state through a violent relaxation phase during which the phase-space density displays a smooth spiral structure presenting a morphology consistent with predictions from self-similar dynamics , ( ii ) a quasi-steady state phase during which radial instabilities can take place at small scales and destroy the spiral structure but do not change quantitatively the properties of the phase-space distribution at the coarse grained level and ( iii ) relaxation to non spherical state due to radial orbit instabilities for n \leq - 1 in the cool case .