  Symmetry defects, e.g., vortices in conventional superfluids, play a critical
role in a complete description of symmetry-breaking phases. In this paper, we
develop the theory of symmetry defects in fractonic superfluids, i.e.,
spontaneously higher-rank symmetry (HRS) breaking phases. By Noether's theorem,
HRS is associated with the conservation law of higher moments, e.g. dipoles,
quadrupoles, and angular moments. We establish finite-temperature phase
diagrams by identifying a series of topological phase transitions via the
renormalization group flow equations and Debye-H\"uckel approximation.
Accordingly, a series of Kosterlitz-Thouless topological transitions are found
to occur successively at different temperatures, which are triggered by
proliferation of defects, defect bound states, and so on. Such a
\emph{hierarchical proliferation} brings rich phase structures. Meanwhile, a
screening effect from sufficiently high density of defect bound states leads to
instability and collapse of the intermediate temperature phases, which further
enriches the phase diagrams. For concreteness, we consider a fractonic
superfluid in which ``angular moments'' are conserved. We then present the
general theory, in which other types of HRS can be analyzed in a similar
manner. Further directions are present at the end of the paper.

Strongly Correlated Electrons;Mesoscale and Nanoscale Physics;High Energy Physics - Theory