Two searches for new phenomena in final states containing a same-flavour opposite-sign lepton (electron or muon) pair, jets, and large missing transverse momentum are presented. These searches make use of proton-proton collision data, collected during 2015 and 2016 at a centre-of-mass energy root s = 13 TeV by the ATLAS detector at the large hadron collider, which correspond to an integrated luminosity of 14.7 fb(-1). Both searches target the pair production of supersymmetric particles, squarks or gluinos, which decay to final states containing a same-flavour opposite-sign lepton pair via one of two mechanisms: a leptonically decaying Z boson in the final state, leading to a peak in the dilepton invariant-mass distribution around the Z boson mass; and decays of neutralinos (e.g.. (chi) over tilde (0)(2) -> l(+)l(-1)(chi) over tilde (0)(1)), yielding a kinematic endpoint in the dilepton invariant-mass spectrum. The data are found to be consistent with the Standard Model expectation. Results are interpreted in simplified models of gluino-pair (squark-pair) production, and provide sensitivity to gluinos (squarks) with masses as large as 1.70 TeV (980 GeV).