Two searches for supersymmetric particles in final states containing a same-flavour opposite-sign lepton pair, jets and large missing transverse momentum are presented. The proton–proton collision data used in these searches were collected at a centre-of-mass energy s=8 TeV by the ATLAS detector at the Large Hadron Collider and corresponds to an integrated luminosity of 20.3 fb -1 . Two leptonic production mechanisms are considered: decays of squarks and gluinos with Z bosons in the final state, resulting in a peak in the dilepton invariant mass distribution around the Z -boson mass; and decays of neutralinos (e.g. χ~20→ℓ+ℓ-χ~10 ), resulting in a kinematic endpoint in the dilepton invariant mass distribution. For the former, an excess of events above the expected Standard Model background is observed, with a significance of three standard deviations. In the latter case, the data are well-described by the expected Standard Model background. The results from each channel are interpreted in the context of several supersymmetric models involving the production of squarks and gluinos.