jats:pWe consider the self-induced motions of three-dimensional oblate spheroids of density jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline1" />jats:tex-math$𝜌s$</jats:tex-math></jats:alternatives></jats:inline-formula> with varying aspect ratios jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline2" />jats:tex-math$AR=b/c⩽1$</jats:tex-math></jats:alternatives></jats:inline-formula>, where jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline3" />jats:tex-math$b$</jats:tex-math></jats:alternatives></jats:inline-formula> and jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline4" />jats:tex-math$c$</jats:tex-math></jats:alternatives></jats:inline-formula> are the spheroids’ centre-pole radius and centre-equator radius, respectively. Vertical motion is imposed on the spheroids such that jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline5" />jats:tex-math$ys(t)=A\sin (2\pi ft)$</jats:tex-math></jats:alternatives></jats:inline-formula> in a fluid of density jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline6" />jats:tex-math$𝜌$</jats:tex-math></jats:alternatives></jats:inline-formula> and kinematic viscosity jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline7" />jats:tex-math$𝜈$</jats:tex-math></jats:alternatives></jats:inline-formula>. As in strictly two-dimensional flows, above a critical value jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline8" />jats:tex-math$ReC$</jats:tex-math></jats:alternatives></jats:inline-formula> of the flapping Reynolds number jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline9" />jats:tex-math$ReA=2Afc/𝜈$</jats:tex-math></jats:alternatives></jats:inline-formula>, the spheroid ultimately propels itself horizontally as a result of fluid–body interactions. For jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline10" />jats:tex-math$ReA$</jats:tex-math></jats:alternatives></jats:inline-formula> sufficiently above jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline11" />jats:tex-math$ReC$</jats:tex-math></jats:alternatives></jats:inline-formula>, the spheroid rapidly settles into a terminal state of constant, unidirectional velocity, consistent with the prediction of Deng jats:italicet al.</jats:italic> (jats:italicPhys. Rev.</jats:italic> E, vol. 94, 2016, 033107) that, at sufficiently high jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline12" />jats:tex-math$ReA$</jats:tex-math></jats:alternatives></jats:inline-formula>, such oscillating spheroids manifest jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline13" />jats:tex-math$m=1$</jats:tex-math></jats:alternatives></jats:inline-formula> asymmetric flow, with characteristic vortical structures conducive to providing unidirectional thrust if the spheroid is free to move horizontally. The speed jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline14" />jats:tex-math$U$</jats:tex-math></jats:alternatives></jats:inline-formula> of propagation increases linearly with the flapping frequency, resulting in a constant Strouhal number jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline15" />jats:tex-math$St(AR)=2Af/U$</jats:tex-math></jats:alternatives></jats:inline-formula>, characterising the locomotive performance of the oblate spheroid, somewhat larger than the equivalent jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline16" />jats:tex-math$St$</jats:tex-math></jats:alternatives></jats:inline-formula> for two-dimensional spheroids, demonstrating that the three-dimensional flow is less efficient at driving locomotion. jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline17" />jats:tex-math$St$</jats:tex-math></jats:alternatives></jats:inline-formula> 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 jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline18" />jats:tex-math$ReA\gtrsim ReC$</jats:tex-math></jats:alternatives></jats:inline-formula>, we observe two distinct types of inherently three-dimensional motion for different aspect ratios. The first, associated with a disk of aspect ratio jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline19" />jats:tex-math$AR=0.1$</jats:tex-math></jats:alternatives></jats:inline-formula> at jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline20" />jats:tex-math$ReA=45$</jats:tex-math></jats:alternatives></jats:inline-formula>, 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 jats:italicet al.</jats:italic> (jats:italicPhys. Rev.</jats:italic> E, vol. 94, 2016, 033107). Such ‘in-phase’ instabilities arise in a relatively narrow band of jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline21" />jats:tex-math$ReA\gtrsim ReC$</jats:tex-math></jats:alternatives></jats:inline-formula>, which band shifts to higher Reynolds number as the aspect ratio increases. (Indeed, for horizontally fixed spheroids with aspect ratio jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline22" />jats:tex-math$AR=0.2$</jats:tex-math></jats:alternatives></jats:inline-formula>, Floquet analysis actually predicts stability at jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline23" />jats:tex-math$ReA=45$</jats:tex-math></jats:alternatives></jats:inline-formula>.) For such a spheroid (jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline24" />jats:tex-math$AR=0.2$</jats:tex-math></jats:alternatives></jats:inline-formula>, jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline25" />jats:tex-math$ReA=45$</jats:tex-math></jats:alternatives></jats:inline-formula>, with sufficiently small mass ratio jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline26" />jats:tex-math$ms/mf=𝜌sVs/(𝜌Vs)$</jats:tex-math></jats:alternatives></jats:inline-formula>, where jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline27" />jats:tex-math$Vs$</jats:tex-math></jats:alternatives></jats:inline-formula> 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 jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112018000629_inline28" />jats:tex-math$QPH$</jats:tex-math></jats:alternatives></jats:inline-formula> quasi-periodic symmetry breaking discussed for sufficiently high aspect ratio two-dimensional elliptical foils in Deng & Caulfield (jats:italicJ. Fluid Mech.</jats:italic>, vol. 787, 2016, pp. 16–49).</jats:p>