First evidence of the B-0 -> J/psi omega decay is found and the B-s(0) -> J/psi eta and B-s(0) -> J/psi eta' decays are studied using a dataset corresponding to an integrated luminosity of 1.0 fb(-1) collected by the LHCb experiment in proton-proton collisions at a centre-of-mass energy of root s = 7 TeV. The branching fractions of these decays are measured relative to that of the B-0 -> J/psi rho(0) decay: B(B-0 -> J/psi omega)/B(B-0 -> J/psi rho(0)) = 0.89 +/- 0.19(stat)(-0.13)(+0.07)(syst), B(B-s(0) -> J/psi eta)/B(B-0 -> J/psi rho(0)) = 14.0 +/- 1.2(stat)(-1.5)(+1.1)(syst)(-1.0)(+1.1)(f(d)/f(s)), B(B-s(0) -> J/psi eta')/B(B-0 -> J/psi rho(0)) = 12.7 +/- 1.1(stat)(-1.3)(+0.5)(syst)(-0.9)(+1.0)(f(d)/f(s)), where the last uncertainty is due to the knowledge of f(d)/f(s), the ratio of b-quark hadronization factors that accounts for the different production rate of B-0 and B-s(0) mesons. The ratio of the branching fractions of B-s(0) -> J/psi eta' and B-s(0) -> J/psi eta decays is measured to be B(B-s(0) -> J/psi eta')/B(B-s(0) -> J/psi eta) = 0.90 +/- 0.09(stat)(-0.02)(+0.06)(syst). (C) 2012 CERN. Published by Elsevier B.V. All rights reserved.