  It is shown that the Goldstone modes associated with a broken continuous
symmetry lead to anomalously large fluctuations of the zero field order
parameter at any temperature below T_c. In dimensions 2<d<4, the variance of
the extensive spontaneous magnetization scales as L^4 with the system size L,
independent of the order parameter dynamics. The anomalous scaling is a
consequence of the 1/q^{4-d} divergence of the longitudinal susceptibility. For
ground states in two dimensions with Goldstone modes vanishing linearly with
momentum, the dynamical susceptibility contains a singular contribution
(q^2-\omega^2/c^2)^{-1/2}. The dynamic structure factor thus exhibits a
critical continuum above the undamped spin wave pole, which may be detected by
neutron scattering in the N\'eel-phase of 2D quantum antiferromagnets.
