Theory and numerical simulations are used to investigate the concentration fluctuations and the microstructure in dilute sedimenting suspensions of orientable and deformable particles at zero Reynolds number. The case of orientable particles is studied using prolate and oblate spheroids, while viscous droplets in the small deformation regime illustrate the effects of deformability. An efficient method based on a point-particle approximation and on smooth localized force representations is implemented to simulate full-scale suspensions with both periodic and slip boundaries, where the latter are used to qualitatively reproduce the effects of horizontal walls. The concentration instability predicted theoretically for suspensions of spheroids is captured in the simulations, and we find that including horizontal walls provides a mechanism for wavenumber selection, in contrast to periodic systems in which the longest wavelength set by the size of the container dominates. A theoretical model for the case of slightly deformable particles is developed, and a linear stability analysis shows that such suspensions are also unstable to concentration fluctuations under sedimentation. In the absence of diffusion, the model predicts that density fluctuations are equally unstable at all wavelengths, but we show that diffusion, whether Brownian or hydrodynamic, should damp high-wavenumber fluctuations. Simulations are also performed for deformable particles, and again an instability is observed that shows a similar mechanism for the wavenumber selection in finite containers. Our results demonstrate that all sedimentation processes of orientable or deformable particles are subject to spontaneous concentration inhomogeneities, which control the sedimentation rates in these systems.