For 1≤α≤2, the family of fast diffusions, ut=[u-αux]x, 0<α<2, coexists with superfast processes which, prior to termination within a finite time, assume a time-space separable form. The remarkable properties of ut[ln(u)]xx guide the understanding of these processes; two interacting kinks form a superfast shrinking pattern or a persisting motion of two poles. For α=1, the superfast axisymmetric diffusion coexists with a modified fundamental diffusion: a response to a singular core and a ring of δ sources. In 3D, fast (0<α<23) and the separable (45≤α<1) superfast processes are distinct. Inhomogeneity of the medium is considered.