  We measure the branching fractions and $CP$ asymmetries for the singly
Cabibbo-suppressed decays $D^{0}\to\pi^{+}\pi^{-}\eta$, $D^{0}\to
K^{+}K^{-}\eta$, and $D^{0}\to\phi\eta$, using 980 fb$^{-1}$ of data from the
Belle experiment at the KEKB $e^+e^-$ collider. We obtain \begin{eqnarray}
\mathcal{B}(D^{0}\to\pi^{+}\pi^{-}\eta) & = & [1.22\pm 0.02\,({\rm stat})\pm
0.02\,({\rm syst})\pm 0.03\,(\mathcal{B}_{\rm ref})]\times 10^{-3}\,, \nonumber
\\ \mathcal{B}(D^{0}\to K^{+}K^{-}\eta) & = & [1.80\,^{+0.07}_{-0.06}\,({\rm
stat})\pm 0.04\,({\rm syst})\pm 0.05\,(\mathcal{B}_{\rm ref})]\times 10^{-4}\,,
\nonumber \\ \mathcal{B}(D^{0}\to\phi\eta) & = & [1.84\pm 0.09\,({\rm stat})\pm
0.06\,({\rm syst})\pm 0.05\,(\mathcal{B}_{\rm ref})]\times 10^{-4}\,, \nonumber
\end{eqnarray} where the third uncertainty ($\mathcal{B}_{\rm ref}$) is from
the uncertainty in the branching fraction of the reference mode $D^{0}\to
K^{-}\pi^{+}\eta$. The color-suppressed decay $D^{0}\to\phi\eta$ is observed
for the first time, with very high significance. The results for the $CP$
asymmetries are \begin{eqnarray} A_{CP}(D^{0}\to\pi^{+}\pi^{-}\eta) & = &
[0.9\pm 1.2\,({\rm stat})\pm 0.5\,({\rm syst})]\%\,, \nonumber \\
A_{CP}(D^{0}\to K^{+}K^{-}\eta) & = & [-1.4\pm 3.3\,({\rm stat})\pm 1.1\,({\rm
syst})]\%\,, \nonumber \\ A_{CP}(D^{0}\to\phi\eta)&= & [-1.9\pm 4.4\,({\rm
stat})\pm 0.6\,({\rm syst})]\%\,. \nonumber \end{eqnarray} The results for
$D^{0}\to\pi^{+}\pi^{-}\eta$ are a significant improvement over previous
results. The branching fraction and $A_{CP}$ results for $D^{0}\to
K^{+}K^{-}\eta$, and the $A_{CP}$ result for $D^{0}\to\phi\eta$, are the first
such measurements. No evidence for $CP$ violation is found in any of these
decays.
