The decays ψ'→γπ(0), γη and γη' are studied using data collected with the BESIII detector at the BEPCII e(+)e(-) collider. The processes ψ'→γπ(0) and ψ'→γη are observed for the first time with signal significances of 4.6σ and 4.3σ, respectively. The branching fractions are determined to be B(ψ'→γπ(0))=(1.58±0.40±0.13)×10(-6), B(ψ'→γη)=(1.38±0.48±0.09)×10(-6), and B(ψ'→γη')=(126±3±8)×10(-6), where the first errors are statistical and the second ones systematic.