  An analysis of $B^+ \to K_{\rm \scriptscriptstyle S}^0 \pi^+$ and $B^+ \to
K_{\rm \scriptscriptstyle S}^0 K^+$ decays is performed with the LHCb
experiment. The $pp$ collision data used correspond to integrated luminosities
of $1{\,fb}^{-1}$ and $2{\,fb}^{-1}$ collected at centre-of-mass energies of
$\sqrt{s}=7\mathrm{\,Te\kern -0.1em V}$ and $\sqrt{s}=8\mathrm{\,Te\kern -0.1em
V}$, respectively. The ratio of branching fractions and the direct {\it CP}
asymmetries are measured to be $\mathcal{B}(B^+ \to K_{\rm \scriptscriptstyle
S}^0 K^+)/\mathcal{B}(B^+ \to K_{\rm \scriptscriptstyle S}^0 \pi^+) = 0.064 \pm
0.009\textrm{(stat.)} \pm 0.004\textrm{(syst.)}$, $\mathcal{A}^{\it CP}(B^+ \to
K_{\rm \scriptscriptstyle S}^0 \pi^+) = -0.022 \pm 0.025\textrm{(stat.)} \pm
0.010\textrm{(syst.)}$ and $\mathcal{A}^{\it CP}(B^+ \to K_{\rm
\scriptscriptstyle S}^0 K^+) = -0.21 \pm 0.14\textrm{(stat.)} \pm
0.01\textrm{(syst.)}$. The data sample taken at $\sqrt{s}=7\mathrm{\,Te\kern
-0.1em V}$ is used to search for $B_c^+ \to K_{\rm \scriptscriptstyle S}^0 K^+$
decays and results in the upper limit $(f_c\cdot\mathcal{B}(B_c^+ \to K_{\rm
\scriptscriptstyle S}^0 K^+))/(f_u\cdot\mathcal{B}(B^+ \to K_{\rm
\scriptscriptstyle S}^0 \pi^+)) < 5.8\times10^{-2}\textrm{at 90% confidence
level}$, where $f_c$ and $f_u$ denote the hadronisation fractions of a
$\bar{b}$ quark into a $B_c^+$ or a $B^+$ meson, respectively.
