  We present a study of rare decay modes $B^{+} \to D_{s}^{+}h^{0}$, $B^{+} \to
D_{s}^{\ast+}h^{0}$, and $B^{+} \to D^{+}h^{0}$, where $h^{0}$ denotes the
neutral mesons $\eta$ or $K^{0}$, using a data sample of $(772 \pm 10 ) \times
10^{6}$ $B\bar{B}$ events produced at the $\Upsilon(4S)$ resonance. The data
were collected by the Belle detector operating at the asymmetric-energy KEKB
collider. We observe no evidence for these decays, so we provide upper limits
at the 90$\%$ confidence level on the branching fractions of $B^{+} \to
D_{s}^{+}h^{0}$, $D_{s}^{\ast+}h^{0}$, and $D^{+}h^{0}$ decay modes. Along with
rare decay modes, we report improved measurements of the color-suppressed decay
branching fractions $\mathcal{B}(\bar{B}^{0} \to D^{0}\eta)$ = (26.6 $\pm$ 1.2
$\pm$ 2.1) $\times$ $10^{-5}$ and $\mathcal{B}(\bar{B}^{0} \to
D^{0}\bar{K}^{0})$ = (5.6 $\pm$ 0.5 $\pm$ 0.2) $\times$ $10^{-5}$. The first
and second uncertainties are statistical and systematic, respectively.
