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 L4 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 Néel phase of 2D quantum antiferromagnets.