We study the Υ(3S,2S)→ηΥ(1S) and Υ(3S,2S)→π^+π^-Υ(1S) transitions with 122×10^6Υ(3S) and 100×10^6Υ(2S) mesons collected by the BABAR detector at the PEP-II asymmetric-energy e^+e^- collider. We measure B[Υ(2S)→ηΥ(1S)]=(2.39±0.31(stat.)±0.14(syst.))×10^(-4) and Γ[Υ(2S)→ηΥ(1S)]/Γ[Υ(2S)→π^+π^-Υ(1S)]=(1.35±0.17(stat.)±0.08(syst.))×10^(-3). We find no evidence for Υ(3S)→ηΥ(1S) and obtain B[Υ(3S)→ηΥ(1S)]<1.0×10^(-4) and Γ[Υ(3S)→ηΥ(1S)]/ Γ[Υ(3S)→π^+π^-Υ(1S)]<2.3×10^(-3) as upper limits at the 90% confidence level. We also provide improved measurements of the Υ(2S)-Υ(1S) and Υ(3S)-Υ(1S) mass differences, 562.170±0.007(stat.)±0.088(syst.) MeV/c^2 and 893.813±0.015(stat.)±0.107(syst.) MeV/c^2, respectively
