We report results of a muon spin relaxation ($mu$SR) study of YFe$_2$Al$_{10}$, a quasi-2D nearly-ferromagnetic metal in which unconventional quantum critical behavior is observed. No static Fe$^{2+}$ magnetism, with or without long-range order, is found down to 19~mK@. The dynamic muon spin relaxation rate~$lambda$ exhibits power-law divergences in temperature and magnetic field, the latter for fields that are too weak to affect the electronic spin dynamics directly. We attribute this to the proportionality of $lambda(omega_mu,T)$ to the dynamic structure factor~$S(omega_mu,T)$, where $omega_mu approx 10^5$--$10^7~mathrm{s}^{-1}$ is the muon Zeeman frequency. These results suggest critical divergences of $S(omega_mu,T)$ in both temperature and frequency. Power-law scaling and a 2D dissipative quantum XY (2D-DQXY) model both yield forms for $S(omega,T)$ that agree with neutron scattering data ($omega approx 10^{12}~mathrm{s}^{-1}$). Extrapolation to $mu$SR frequencies agrees semi-quantitatively with the observed temperature dependence of $lambda(omega_mu,T)$, but predicts frequency independence for $omega_mu ll T$ in extreme disagreement with experiment. We conclude that the quantum critical spin dynamics of YFe$_2$Al$_{10}$ are not well understood at low frequencies.
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