By analyzing 7.33 fb^{-1} of e^{+}e^{-} annihilation data collected at center-of-mass energies between 4.128 and 4.226 GeV with the BESIII detector, we report the observation of the semileptonic decay D_{s}^{+}→η^{'}μ^{+}ν_{μ}, with a statistical significance larger than 10σ, and the measurements of the D_{s}^{+}→ημ^{+}ν_{μ} and D_{s}^{+}→η^{'}μ^{+}ν_{μ} decay dynamics for the first time. The branching fractions of D_{s}^{+}→ημ^{+}ν_{μ} and D_{s}^{+}→η^{'}μ^{+}ν_{μ} are determined to be (2.235±0.051_{stat}±0.052_{syst})% and (0.801±0.055_{stat}±0.028_{syst})%, respectively, with precision improved by factors of 6.0 and 6.6 compared to the previous best measurements. Combined with the results for the decays D_{s}^{+}→ηe^{+}ν_{e} and D_{s}^{+}→η^{'}e^{+}ν_{e}, the ratios of the decay widths are examined both inclusively and in several ℓ^{+}ν_{ℓ} four-momentum transfer ranges. No evidence for lepton flavor universality violation is found within the current statistics. The products of the hadronic form factors f_{+,0}^{η^{(')}}(0) and the c→s Cabibbo-Kobayashi-Maskawa matrix element |V_{cs}| are determined. The results based on the two-parameter series expansion are f_{+,0}^{η}(0)|V_{cs}|=0.452±0.010_{stat}±0.007_{syst} and f_{+,0}^{η^{'}}(0)|V_{cs}|=0.504±0.037_{stat}±0.012_{syst}, which help to constrain present models on f_{+,0}^{η^{(')}}(0). The forward-backward asymmetries are determined to be ⟨A_{FB}^{η}⟩=-0.059±0.031_{stat}±0.005_{syst} and ⟨A_{FB}^{η^{'}}⟩=-0.064±0.079_{stat}±0.006_{syst} for the first time, which are consistent with the theoretical calculation.