Uniform arrays of particles tend to cluster as they sediment in viscous fluids. Shape anisotropy of the particles enriches this dynamics by modifying the mode structure and the resulting instabilities of the array. A one-dimensional lattice of sedimenting spheroids in the Stokesian regime displays either an exponential or an algebraic rate of clustering depending on the initial lattice spacing (Chajwa et al. 2020 Phys. Rev. X vol. 10, pp. 041016). This is caused by an interplay between the Crowley mechanism, which promotes clumping, and a shape-induced drift mechanism, which subdues it. We theoretically and experimentally investigate the sedimentation dynamics of one-dimensional lattices of oblate spheroids or discs and show a stark difference in clustering behaviour: the Crowley mechanism results in clumps comprising several spheroids, whereas the drift mechanism results in pairs of spheroids whose asymptotic behaviour is determined by pair–hydrodynamic interactions. We find that a Stokeslet, or point-particle, approximation is insufficient to accurately describe the instability and that the corrections provided by the first reflection are necessary for obtaining some crucial dynamical features. As opposed to a sharp boundary between exponential growth and neutral eigenvalues under the Stokeslet approximation, the first-reflection correction leads to exponential growth for all initial perturbations, but far more rapid algebraic growth than exponential growth at large dimensionless lattice spacing $\tilde {d}$. For discs with aspect ratio $0.125$, corresponding to the experimental value, the instability growth rate is found to decrease with increasing lattice spacing $\tilde {d}$, approximately as $\tilde {d}^{ -4.5}$, which is faster than the $\tilde {d}^{-2}$ for spheres (Crowley 1971 J. Fluid Mech. vol. 45, pp. 151–159). It is shown that the first-reflection correction has a stabilising effect for small lattice spacing and a destabilising effect for large lattice spacing. Sedimenting pairs predominantly come together to form an inverted ‘T’, or ‘$\perp$’, which our theory accounts for through an analysis that builds on Koch & Shaqfeh (1989 J. Fluid Mech. vol. 209, pp. 521–542). This structure remains stable for a significant amount of time.