Measurements have been made of the recoilless fraction and broadening of Mössbauer γ rays Rayleigh scattered from glycerol and sec-butyl benzene in the supercooled liquid, glassy and polycrystalline states, and from polycrystalline benzene. It is concluded that the vibration amplitude (possibly rotational in origin) in the supercooled liquids can be considerably larger than in the corresponding polycrystals at the same temperature. Energy broadening due to molecular diffusion was observed in supercooled liquid glycerol, in approximate agreement with what would be expected from viscosity values and a naïve application of the Einstein formula. No such broadening was observed in the scattered radiation from sec-butyl benzene, contrary to that expected from viscosity data. With polycrystalline benzene no broadening from the rotational motion deduced from nuclear magnetic resonance experiments was seen, and this is interpreted as evidence that the molecules rotate in discrete jumps, such that the spatial positions of a molecule both before and after a jump are the same, rather than continuously or as a rotational Brownian motion.