  We study theoretically spin decoherence and intrinsic spin noise in
semiconductor quantum wires caused by an interplay of electron hopping between
the localized states and the hyperfine interaction of electron and nuclear
spins. At a sufficiently low density of localization sites the hopping rates
have an exponentially broad distribution. It allows the description of the spin
dynamics in terms of closely-situated "pairs" of sites and single "reaching"
states, from which the series of hops result in the electron localized inside a
"pair". The developed analytical model and numerical simulations demonstrate
disorder-dependent algebraic tails in the spin decay and power-law
singularity-like features in the low-frequency part of the spin noise spectrum.
