We present the results of searches for B decays to charmless two-body final states containing eta(') or omega mesons, based on 20.7 fb(-1) of data collected with the BABAR detector. We find the branching fractions Beta(B(+)-->eta(')K(+)) = (70+/-8+/-5) x 10(-6), Beta(B(0)-->eta(')K(0)) = (42(+13)(-11) +/- 4) x 10(-6), and Beta(B(+)-->omega pi(+)) = (6.6(+2.1)(-1.8) +/- 0.7) x 10(-6), where the first error quoted is statistical and the second is systematic. We give measurements of four additional modes for which the 90% confidence level upper limits are Beta(B(+)-->eta(')pi(+)) < 12 x 10(-6), Beta(B(+)-->omega K(+)) < 4 x 10(-6), Beta(B(0)-->omega K(0)) < 13 x 10(-6), and Beta(B(0)-->omega pi(0)) < 3 x 10(-6).