  We address the nature of spin dynamics in various integrable and
non-integrable, isotropic and anisotropic quantum spin-$S$ chains, beyond the
paradigmatic $S=1/2$ Heisenberg model. In particular, we investigate the
algebraic long-time decay $\propto t^{-1/z}$ of the spin-spin correlation
function at infinite temperature, using state-of-the-art simulations based on
tensor network methods. We identify three universal regimes for the spin
transport, independent of the exact microscopic model: (i) superdiffusive with
$z=3/2$, as in the Kardar-Parisi-Zhang universality class, when the model is
integrable with extra symmetries such as spin isotropy that drive the Drude
weight to zero, (ii) ballistic with $z=1$ when the model is integrable with a
finite Drude weight, and (iii) diffusive with $z=2$ with easy-axis anisotropy
or without integrability, at variance with previous observations.
