We study the processes e+e-→γISRJ/ψ, where J/ψ→π+π-π0, J/ψ→K+K-π0, and J/ψ→KS0K±π∓ using a data sample of 519 fb-1 recorded with the BABAR detector operating at the SLAC PEP-II asymmetric-energy e+e- collider at center-of-mass energies at and near the ϒ(nS) (n=2,3,4) resonances. We measure the ratio of branching fractions R1=B(J/ψ→K+K-π0)B(J/ψ→π+π-π0) and R2=B(J/ψ→KS0K±π∓)B(J/ψ→π+π-π0). We perform Dalitz plot analyses of the three J/ψ decay modes and measure fractions for resonances contributing to the decays. We also analyze the J/ψ→π+π-π0 decay using the Veneziano model. We observe structures compatible with the presence of ρ(1450) in all three J/ψ decay modes and measure the relative branching fraction: R(ρ(1450))=B(ρ(1450)→K+K-)B(ρ(1450)→π+π-)=0.307±0.084(stat)±0.082(sys).