  Using an $e^{+}e^{-}$ annihilation data sample corresponding to an integrated
luminosity of $3.19~\mathrm{fb}^{-1}$ and collected at a center-of-mass energy
$\sqrt{s} = 4.178~\mathrm{GeV}$ with the BESIII detector, we measure the
absolute branching fractions $\mathcal{B}(D_{s}^{+} \rightarrow K_{S}^{0}K^{+})
= (1.425\pm0.038_{\rm stat.}\pm0.031_{\rm syst.})\%$ and $\mathcal{B}(D_{s}^{+}
\rightarrow K_{L}^{0}K^{+}) =(1.485\pm0.039_{\rm stat.}\pm0.046_{\rm
syst.})\%$. The branching fraction of $D_{s}^{+} \rightarrow K_{S}^{0}K^{+}$ is
compatible with the world average and that of $D_{s}^{+} \rightarrow
K_{L}^{0}K^{+}$ is measured for the first time. We present the first
measurement of the $K_{S}^{0}$-$K_{L}^{0}$ asymmetry in the decays $D_{s}^{+}
\rightarrow K_{S,L}^{0}K^{+}$, and $R(D_{s}^{+} \rightarrow
K_{S,L}^{0}K^{+})=\frac{\mathcal{B}(D_{s}^{+} \rightarrow K_{S}^{0}K^{+})
-\mathcal{B}(D_{s}^{+} \rightarrow K_{L}^{0}K^{+})}{\mathcal{B}(D_{s}^{+}
\rightarrow K_{S}^{0}K^{+}) +\mathcal{B}(D_{s}^{+} \rightarrow
K_{L}^{0}K^{+})}= (-2.1\pm1.9_{\rm stat.}\pm1.6_{\rm syst.})\%$. In addition,
we measure the direct $CP$ asymmetries $A_{\rm CP}(D_{s}^{\pm} \rightarrow
K_{S}^{0}K^{\pm}) = (0.6\pm2.8_{\rm stat.}\pm0.6_{\rm syst.})\%$ and $A_{\rm
CP}(D_{s}^{\pm} \rightarrow K_{L}^{0}K^{\pm}) = (-1.1\pm2.6_{\rm
stat.}\pm0.6_{\rm syst.})\%$.
