Optical imaging polarimetry has been performed on a small sample of objects which are associated with that stage of stellar evolution occurring between the Asymptotic Giant Branch and full Planetary Nebula. Three such systems are considered, specifically, the young planetary nebulae M 1-16 and Mz3, and the protoplanetary nebula, IRAS 09371+1212 (the 'Frosty Leo' nebula). The work is based upon CCD polarimetry obtained with the Durham Imaging Polarimeter. Planetary nebulae are believed to form as a low to intermediate mass star evolves from the main sequence, through the mass-loss stages of the Red Giant Branch and Asymptotic Giant Branch, towards its final destiny as a White Dwarf. A brief review of the relevant aspects of post-main sequence stellar evolution is given as a basis for understanding the transitionary planetary nebula phase in relation to the character of the central star and its role in the creation of a nebula. The theory of light scattering from both homogeneous and core-mantle spherical dust grains (Mie theory) is discussed. The results of a series of scattering calculations, using the theory, for dust grains composed of those materials believed to be abundant in the atmospheres of late-type stars and planetary nebulae are presented. The levels of polarization and scattered intensities predicted in the scattering analysis have been applied in the interpretation of the polarimetric data for the three planetary nebulae. Constraints upon the nature of the dust component, the size distribution of the dust and the nebula geometry are suggested for each of the nebulae. The inferred character of the dust material is in good agreement with the classification of the nebulae using the two-colour diagram for the IRAS fluxes.