First evidence of the B0→J/ψω decay is found and the Bs0→J/ψη and Bs0→J/ψη′ 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 s=7 TeV. The branching fractions of these decays are measured relative to that of the B0→J/ψρ0 decay:B(B0→J/ψω)B(B0→J/ψρ0)=0.89±0.19(stat)−0.13+0.07(syst),B(Bs0→J/ψη)B(B0→J/ψρ0)=14.0±1.2(stat)−1.5+1.1(syst)−1.0+1.1(fdfs),B(Bs0→J/ψη′)B(B0→J/ψρ0)=12.7±1.1(stat)−1.3+0.5(syst)−0.9+1.0(fdfs), where the last uncertainty is due to the knowledge of fd/fs, the ratio of b-quark hadronization factors that accounts for the different production rate of B0 and Bs0 mesons. The ratio of the branching fractions of Bs0→J/ψη′ and Bs0→J/ψη decays is measured to beB(Bs0→J/ψη′)B(Bs0→J/ψη)=0.90±0.09(stat)−0.02+0.06(syst).