  We study the spin transport properties of some disordered spin chains with a
special focus on the distribution of the frequency-dependent spin conductivity.
In the cases of interest here, the systems are governed by an effectively
infinite disorder at low energies. A hallmark of this behavior is the wide
discrepancy between the average and the typical values of some physical
quantities, which are described by extremely broad distributions. We show that
such is also the case of the spin conductivity, whose average value is metallic
but whose typical one, the physically relevant quantity, is insulating. This
solves the apparent contradiction between the prediction of a spin metallic
phase of the spin-1/2 disordered XX chain and its known localized behavior
(after a mapping to free fermions). Our results are based on analytical and
numerical implementations of a strong-disorder renormalization group as well as
exact diagonalization studies. We present our analyses in very general terms,
valid for systems of any spin $S$ value, but the cases of $S=1/2$ and 1 are
studied in greater detail.
