Emulsion droplets, composed of long linear molecules (e.g. n-alkanes) and stabilized by long chain surfactants, can spontaneously change their shape upon cooling and morph from spheres into regular polyhedra, polygonal platelets, rods and thin fibers. Until now, these dynamic shape changes have been studied upon continuous temperature decrease. Therefore, it had remained unclear whether the observed shapes were stable or they appeared as intermediate states during the transformation of the initial sphere into some final shape(s). In the current study, we performed dedicated experiments at fixed temperature to distinguish between the stable, metastable and unstable shapes and to define the factors which affect the kinetics of the observed transitions. The results showed that the stable shapes are equilateral triangular platelets (in all surfactant solutions) and the rod-like particles, when stabilized by surfactants with narrow compositional variations. The distorted octahedrons (seen as hexagonal platelets in transmitted light) were metastable and preserved their shape for many minutes. The other shapes, e.g. the flat hexagonal and tetragonal platelets, were unstable and continued to transform until they reached one of the stable states or alternatively – until they formed spheroidal droplets extruding thin fibers from their bodies. These experimental results are in general agreement with the theoretical model developed by Haas et al. (Phys. Rev. Lett. 2017, Phys. Rev. Res. 2019). The rate of shape transformations increased with the level of subcooling and/or with the decrease of drop size.