Several similarity laws for the collisionless interaction of ultra-intense electromagnetic fields with a plasma of an arbitrary initial shape are presented. Both ultra-relativistic and non-relativistic cases are covered. The ion motion is included. A relation to the S-similarity described in Pukhov et al (2004 Plasma Phys. Control. Fusion46 B179) and Gordienko and Pukhov (2005 Phys. Plasmas12 043109) is established. A brief discussion of possible ways of experimental verification of scaling laws is presented. The results can be of interest for experiments and numerical simulations in the areas of ion acceleration, harmonic generation, magnetic field generation and Coulomb explosion of clusters.