Nanostructures, such as quantum dots, nanocrystals and nanowires, made of wurtzite ZnO have recently attracted attention due to their proposed applications in optoelectronic devices. Raman spectroscopy has been widely used to study the optical phonon spectrum modification in ZnO nanostructures as compared to bulk crystals. Understanding the phonon spectrum change in wurtzite nanostructures is important because the optical phonons affect the light emission and absorption. The interpretation of the phonon peaks in the Raman spectrum from ZnO nanostructures continues to be the subject of debates. Here we present a comparative study of micro-Raman spectra from ZnO quantum dots, nanocrystals and nanowires. Several possible mechanisms for the peak position shifts, i.e., optical phonon confinement, phonon localization on defects and laser-induced heating, are discussed in details. We show that the shifts of similar to 2 cm(-1) in non-Resonant spectra are likely due to the optical phonon confinement in ZnO quantum dots with the average diameter of 4 nm. The small shifts in the non-Resonant spectra from ZnO nanowires with the diameter similar to 20 nm - 50 nm can be attributed to either defects or large size dispersion, which results in a substantial contribution from nanowires with smaller diameters. The large red-shifts of similar to 10 cm(-1) in the resonant Raman spectrum from nanocrystals were proved to be due to local laser heating.