  Anomalous diffusion phenomena occur on length scales spanning from
intracellular to astrophysical ranges. A specific form of decay at large
argument of the probability density function of rescaled displacement (scaling
function) is derived and shown to imply universal singularities in the
normalized cumulant generator. Exact calculations for continuous time random
walks provide paradigmatic examples connected with singularities of second
order phase transitions. In the biased case scaling is restricted to
displacements in the drift direction and singularities have no equilibrium
analogue.
