  In the easy-plane regime of XXZ spin chains, spin transport is ballistic,
with a Drude weight that has a discontinuous fractal dependence on the value of
the anisotropy $\Delta = \cos \pi \lambda$ at nonzero temperatures. We show
that this structure necessarily implies the divergence of the low-frequency
conductivity for generic irrational values of $\lambda$. Within the framework
of generalized hydrodynamics, we show that in the high-temperature limit the
low-frequency conductivity at a generic anisotropy scales as $\sigma(\omega)
\sim 1/\sqrt{\omega}$; anomalous response occurs because quasiparticles undergo
L\'evy flights. For rational values of $\lambda$, the divergence is cut off at
low frequencies and the corrections to ballistic spin transport are diffusive.
We also use our approach to recover that at the isotropic point $\Delta=1$,
spin transport is superdiffusive with $\sigma(\omega) \sim \omega^{-1/3}$. We
support our results with extensive numerical studies using matrix-product
operator methods.
