We consider the self-induced motions of three-dimensional oblate spheroids of density \unicode[STIX]{x1D70C}_{s} with varying aspect ratios AR=b/c⩽1, where b and c are the spheroids’ centre-pole radius and centre-equator radius, respectively. Vertical motion is imposed on the spheroids such that y_{s}(t)=A\sin (2\unicode[STIX]{x03C0}ft) in a fluid of density \unicode[STIX]{x1D70C} and kinematic viscosity \unicode[STIX]{x1D708}. As in strictly two-dimensional flows, above a critical value ReC of the flapping Reynolds number Re_{A}=2Afc/\unicode[STIX]{x1D708}, the spheroid ultimately propels itself horizontally as a result of fluid–body interactions. For ReA sufficiently above ReC, the spheroid rapidly settles into a terminal state of constant, unidirectional velocity, consistent with the prediction of Deng et al. (Phys. Rev. E, vol. 94, 2016, 033107) that, at sufficiently high ReA, such oscillating spheroids manifest m=1 asymmetric flow, with characteristic vortical structures conducive to providing unidirectional thrust if the spheroid is free to move horizontally. The speed U of propagation increases linearly with the flapping frequency, resulting in a constant Strouhal number St(AR)=2Af/U, characterising the locomotive performance of the oblate spheroid, somewhat larger than the equivalent St for two-dimensional spheroids, demonstrating that the three-dimensional flow is less efficient at driving locomotion. St decreases with increasing aspect ratio for both two-dimensional and three-dimensional flows, although the relative disparity (and hence relative inefficiency of three-dimensional motion) decreases. For flows with ReA≳ReC, we observe two distinct types of inherently three-dimensional motion for different aspect ratios. The first, associated with a disk of aspect ratio AR=0.1 at ReA=45, consists of a ‘stair-step’ trajectory. This trajectory can be understood through consideration of relatively high azimuthal wavenumber instabilities of interacting vortex rings, characterised by in-phase vortical structures above and below an oscillating spheroid, recently calculated using Floquet analysis by Deng et al. (Phys. Rev. E, vol. 94, 2016, 033107). Such ‘in-phase’ instabilities arise in a relatively narrow band of ReA≳ReC, which band shifts to higher Reynolds number as the aspect ratio increases. (Indeed, for horizontally fixed spheroids with aspect ratio AR=0.2, Floquet analysis actually predicts stability at ReA=45.) For such a spheroid (AR=0.2, ReA=45, with sufficiently small mass ratio m_{s}/m_{f}=\unicode[STIX]{x1D70C}_{s}V_{s}/(\unicode[STIX]{x1D70C}V_{s}), where Vs is the volume of the spheroid) which is free to move horizontally, the second type of three-dimensional motion is observed, initially taking the form of a ‘snaking’ trajectory with long quasi-periodic sweeping oscillations before locking into an approximately elliptical ‘orbit’, apparently manifesting a three-dimensional generalisation of the QPH quasi-periodic symmetry breaking discussed for sufficiently high aspect ratio two-dimensional elliptical foils in Deng & Caulfield (J. Fluid Mech., vol. 787, 2016, pp. 16–49).</jats:p