  Multipole symmetries are of interest both as a window on fracton physics and
as a crucial ingredient in realizing new universality classes for quantum
dynamics. Here we address the question of whether and when multipole symmetries
can be spontaneously broken, both in thermal equilibrium and at zero
temperature. We derive generalized Mermin-Wagner arguments for the total or
partial breaking of multipolar symmetry groups and generalized Imry-Ma
arguments for the robustness of such multipolar symmetry breaking to disorder.
We present both general results and explicit examples. Our results should be
directly applicable to quantum dynamics with multipolar symmetries and also
provide a useful stepping stone to understanding the robustness of fracton
phases to thermal fluctuations, quantum fluctuations, and disorder.
