Modulational instability is a phenomenon that reveals itself as the exponential growth of weak perturbations in the presence of an intense pump beam propagating in a nonlinear medium. It plays a key role in such nonlinear optical processes as supercontinuum generation, light filamentation, rogue waves, and ring (or necklace) beam formation. To date, a majority of studies of these phenomena have focused on light-matter interactions in self-focusing Kerr media existing in nature. However, a large and tunable nonlinear response of a colloidal suspension can be tailored at will by judiciously engineering the optical polarizability. Here, we analytically and numerically show the possibility of necklace beam generation originating from spatial modulational instability of vortex beams in engineered soft-matter nonlinear media with different types of exponential nonlinearity.