Observation of B0-->omega K0, B+-->eta pi+, and B+-->eta K+ and study of related decays.

We present measurements of branching fractions and charge asymmetries for seven B-meson decays with an eta, eta', or omega meson in the final state. The data sample corresponds to 89x10(6) BB pairs produced from e(+)e(-) annihilation at the Upsilon(4S) resonance. We measure the following branching fractions in units of 10(-6): B(B+-->eta pi(+))=5.3+/-1.0+/-0.3, B(B+-->eta K+)=3.4+/-0.8+/-0.2, B(B0-->eta K0)=2.9+/-1.0+/-0.2 (<5.2, 90% C.L.), B(B+-->eta(')pi(+))=2.7+/-1.2+/-0.3 (<4.5, 90% C.L.), B(B+-->omega pi(+))=5.5+/-0.9+/-0.5, B(B+-->omega K+)=4.8+/-0.8+/-0.4, and B(B0-->omega K0)=5.9(+1.6)(-1.3)+/-0.5. The charge asymmetries are A(ch)(B+-->eta pi(+))=-0.44+/-0.18+/-0.01, A(ch)(B+-->eta K+)=-0.52+/-0.24+/-0.01, A(ch)(B+-->omega pi(+))=0.03+/-0.16+/-0.01, and A(ch)(B+-->omega K+)=-0.09+/-0.17+/-0.01.