  Higher-form symmetries are a valuable tool for classifying topological phases
of matter. However, emergent higher-form symmetries in interacting many-body
quantum systems are not typically exact due to the presence of topological
defects. In this paper, we develop a systematic framework for building
effective theories with approximate higher-form symmetries, i.e. higher-form
symmetries that are weakly explicitly broken. We focus on a continuous U(1)
q-form symmetry and study various patterns of symmetry breaking. This includes
spontaneous or explicit breaking of higher-form symmetries, as well as
pseudo-spontaneous symmetry breaking patterns where the higher-form symmetry is
both spontaneously and explicitly broken. We uncover a web of dualities between
such phases and highlight their role in describing the presence of dynamical
higher-form vortices. In order to study the out-of-equilibrium dynamics of
these phases of matter, we formulate respective hydrodynamic theories and study
the spectra of excitations exhibiting higher-form charge relaxation and
Goldstone relaxation effects. We show that our framework is able to describe
various phase transitions due to proliferation of vortices or defects. This
includes the melting transition in smectic crystals, the plasma phase
transition from polarised gases to magnetohydrodynamics, the spin-ice
transition, the superfluid to neutral fluid transition and the Meissner effect
in superconductors, among many others.
