  We investigate in this work the spectrum of singularities of super-Brownian
motion with stable branching. The main purpose is to provide a uniform
description of the latter in high dimension $d\geq\tfrac{2}{\gamma-1}$,
presenting the singularities existing at every time t and characterising as
well the set of random times at which singularities of higher order appear. In
lower dimensions, we give a partial description of the singularities which
complement the recent study of the density by Mytnik and Wachtel.
