We measure the branching fractions and CP asymmetries for the singly Cabibbo-suppressed decays D-0 -> pi(+)pi(-)eta, D-0 -> K+K-eta, and D-0 -> phi eta, using 980 fb(-1) of data from the Belle experiment at the KEKB e(+)e(-) collider. We obtain B(D-0 -> pi+pi-eta) = [1.22 +/- 0.02 (stat) +/- 0.02 (syst) +/- 0.03 (B-ref)] x 10(-3), B(D-0 -> K+K-eta) = [1.801(-0.06)(+0.07) (stat) +/- 0.04 (syst) +/- 0.05 (B-ref)] x 10(-4), B(D-0 -> phi eta) = [1.84 +/- 0.09 (stat) +/- 0.06 (syst) +/- 0.05 (B-ref)] x 10(-4), where the third uncertainty (B-ref) is from the uncertainty in the branching fraction of the reference mode D-0 -> K-pi(+)eta.
The color-suppressed decay D-0 -> phi eta is observed for the first time, with very high significance. The results for the CP asymmetries are A(CP) (D-0 -> pi(+)pi(-)eta) = [0.9 +/- 1.2 (stat) +/- 0.5 (syst)]%, A(CP)(D-0 -> K+K-eta)=[-1.4 +/- 3.3 (stat) +/- 1.1 (syst)]%, A(CP)(D-0 -> phi eta) = [-1.9 +/- 4.4 (stat) +/- 0.6 (syst)]%.
The results for D-0 -> pi(+)pi(-)eta are a significant improvement over previous results. The branching fraction and Acp results for D-0 -> K+K-eta, and the A(CP) result for D-0 -> phi eta, are the first such measurements.
No evidence for CP violation is found in any of these decays.