  We extend and refine a general method to extract the multipole moments of
arbitrary stationary spacetimes and apply it to the study of a large family of
regular horizonless solutions to $ {\cal N}{\,=\,}2$ four-dimensional
supergravity coupled to four Abelian gauge fields. These microstate geometries
can carry angular momentum and have a much richer multipolar structure than the
Kerr black hole. In particular they break the axial and equatorial symmetry,
giving rise to a large number of nontrivial multipole moments. After studying
some analytical examples, we explore the four-dimensional parameter space of
this family with a statistical analysis. We find that microstate mass and spin
multipole moments are typically (but not always) larger that those of a Kerr
black hole with the same mass and angular momentum. Furthermore, we find
numerical evidence that some invariants associated with the (dimensionless)
moments of these microstates grow monotonically with the microstate size and
display a global minimum at the black-hole limit, obtained when all centers
collide. Our analysis is relevant in the context of measurements of the
multipole moments of dark compact objects with electromagnetic and
gravitational-wave probes, and for observational tests to distinguish fuzzballs
from classical black holes.
