We report a search for CP violation in the decay modes $D^{\pm} \to K^0_{\scriptscriptstyle S} K^\pm$, $D_s^{\pm} \to K^0_{\scriptscriptstyle S} K^\pm$, and $D_s^{\pm}\to K^0_{\scriptscriptstyle S} \pi^\pm$ using a data set corresponding to an integrated luminosity of $469\,{\,fb}^{-1}$ collected with the {\slshape B\kern-0.1em{\smaller A}\kern-0.1em B\kern-0.1em{\smaller A\kern-0.2em R}} detector at the PEP-II asymmetric energy $e^+e^-$ storage rings. The decay rate CP asymmetries, $A_{CP}$, are determined to be $(+0.13 \pm 0.36 \stat \pm 0.25 \syst)%$, $(-0.05 \pm 0.23 \stat \pm 0.24 \syst)%$, and $(+0.6 \pm 2.0 \stat \pm 0.3 \syst)%$, respectively. These measurements are consistent with zero, and also with the standard model prediction ($(-0.332 \pm 0.006)%$ for the $D^{\pm} \to K^0_{\scriptscriptstyle S} K^\pm$ and $D_s^{\pm} \to K^0_{\scriptscriptstyle S} K^\pm$ modes, and $(+0.332 \pm 0.006)%$ for the $D_s^{\pm}\to K^0_{\scriptscriptstyle S} \pi^\pm$ mode). They are the most precise determinations to date.