Using a sample of 1.31 x 10(9) J/psi events collected with the BESIII detector at the BEPCII collider, the decays J/psi -> phi pi(+)pi(-)pi(0) and J/psi -> phi pi(0)pi(0)pi(0) are investigated. The isospin- violating decay J/psi -> phi pi(0)f(0)(980) with f(0)(980)-> pi pi is observed for the first time. The width of the f(0)(980) obtained from the dipion mass spectrum is found to be much smaller than the world average value. In the pi(0)f(0)(980) mass spectrum, there is evidence of f(1)(1285) production. By studying the decay J/psi ->eta', the branching fractions of eta' -> pi(+)pi(-)pi(0) and eta' -> pi(0)pi(0)pi(0), as well as their ratio, are also measured.