Branching fractions and $CP$-violating asymmetries of charmless ${\overline{B}}_s^0\to PPP$ decays with $P=K$, $\pi$ are studied using a simple model based on the factorization approach. The penguin-dominated modes ${\overline{B}}_s^0\to {\overline{K}}^0{K}^{+}{\pi }^{-}$ and $K^0{K}^{-}{\pi }^{+}$ have the largest rates among the three-body $B_s$ decays, and they are dominated by the $K_0^{*}(1430)$ resonances and nonresonant contributions. The branching fraction of ${\overline{B}}_s^0\to K^0{K}^{+}{K}^{-}$ is predicted to be of order $1.4\times {10}^{-6}$, which is in the middle of the measured result $(0.2–3.4)\times {10}^{-6}$ obtained by LHCb. We find sizable $CP$ violation in $K^0{\pi }^{+}{\pi }^{-}$, $K^0{\pi }^0{\pi }^0$, $K^0{K}^{+}{K}^{-}$, and $K_S{K}_S{K}_S$ channels. Just as the $B_u$ sector, the $CP$ asymmetries of ${\overline{B}}_s^0\to K^0{\pi }^{+}{\pi }^{-}$ and ${\overline{B}}_s^0\to K^0{K}^{+}{K}^{-}$ have similar magnitudes but are opposite in sign. Several model-independent relations based on $U$-spin symmetry for $CP$ asymmetries are derived. Although they cannot be tested by the present available data, they can be checked by the dynamical calculations. Since the $U$-spin symmetry that relates various matrix elements of scalar densities, for example, $⟨K^{0}K|\overline{d}q|0⟩$ and $⟨{\overline{K}}^0\pi |\overline{s}q|0⟩$, is badly broken, the derived $U$-spin relations are generally not well respected.

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