The polarized optical absorption spectra of different quaterthiophene single crystals in the energy region of the exciton bands originating from the first molecular transition are reported as measured in the temperatures ranging from 7 to 140 K. The intrinsic higher mobility of the b-polarized 0-0 a(u) exciton both with respect to its replicas and to the a-polarized structures is demonstrated in high quality crystals at the lowest temperatures. The influence of structural disorder on mobility is discussed considering, for the different samples, the measured lineshape and linewidth of the absorption peaks, and the relative lineshift and intensity ratio between the 0-0 a(u) line and its first replica at the lowest temperature. The influence of dynamic disorder is discussed considering the lineshape and linewidth of the measured peaks as a function of temperature for both polarizations in the framework of the exciton-phonon coupling theory.