We present Raman scattering measurements on high-quality freestanding AlN single crystals. Polarization studies provide clear identification of all allowed phonons. We report the smallest phonon linewidths observed in AlN, which provide a direct measurement of the anharmonic decay of phonons in this material. Also from the Raman mode frequencies and reported index of refraction values we estimated the ordinary and extraordinary dielectric constant values. The calculated values are considerably different from previously reported values of the dielectric constant of AlN.