The interplay between the nematic order and magnetism in FeSe is not yet well understood. There is a controversy concerning the relationship between orbital and spin degrees of freedom in FeSe and their relevance for superconductivity. Here we investigate the effect of S substitution on the nematic transition temperature ($T_{rm n}$) and the low-energy spin fluctuations (SF) in FeSe single crystals. We show that the low-energy SF emerge below the nematic transition. The difference between the onset temperature for the critical SF ($T_{rm SF}$) and $T_{rm n}$ is small for FeSe but significantly increases with S substitution. Below $T_{rm SF}$ the Korringa relation is violated and the effective muon hyperfine coupling constant changes a sign. Our results exclude a direct coupling of the low-energy SF to the electronic nematic order indicating a presence of multiple spin degrees of freedom in FeSe$_{rm 1-x}$S$_{rm x}$.
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