  Speckle patterns produced by random optical fields with two (or more) widely
different correlation lengths exhibit speckle spots that are themselves highly
speckled. Using computer simulations and analytic theory we present results for
the point singularities of speckled speckle fields: optical vortices in scalar
(one polarization component) fields; C points in vector (two polarization
component) fields. In single correlation length fields both types of
singularities tend to be more{}-or{}-less uniformly distributed. In contrast,
the singularity structure of speckled speckle is anomalous: for some sets of
source parameters vortices and C points tend to form widely separated giant
clusters, for other parameter sets these singularities tend to form chains that
surround large empty regions. The critical point statistics of speckled speckle
is also anomalous. In scalar (vector) single correlation length fields phase
(azimuthal) extrema are always outnumbered by vortices (C points). In contrast,
in speckled speckle fields, phase extrema can outnumber vortices, and azimuthal
extrema can outnumber C points, by factors that can easily exceed $10^{4}$ for
experimentally realistic source parameters.
