  We analyze the equilibrium frequency-dependent spin current noise and spin
conductance through a quantum dot in the local moment regime. Spin current
correlations behave markedly differently from charge correlations. Equilibrium
spin correlations are characterized by two universal scaling functions in the
absence of an external field: one of them is related to charge correlations,
while the other one describes cross-spin correlations. We characterize these
functions using a combination of perturbative and non-perturbative methods. We
find that at low temperatures spin cross-correlations are suppressed at
frequencies below the Kondo scale, $T_K$, and a dynamical spin accumulation
resonance is found at the Kondo energy, $\omega \sim T_K$. At higher
temperatures, $T>T_K$, surprising low-frequency anomalies related to overall
spin conservation appear in the spin noise and spin conductance, and the
Korringa rate is shown to play a distinguished role. The transient spin current
response also displays universal and singular properties.
