Fluctuations in photon number and quantum phase and in quadratures of the displaced Kerr states have been studied for different values of Kerr evolution parameter. The dynamics of phase distributions, taking into account that of the preferred phase, have resulted in well centered peaks in the phase interval. The role of the preferred phase in the distributions of quadratures has been considered, leading to approximate identification of conjugate quadratures with the number-phase-operator pair. Nonclassical properties have been demonstrated in the case of noise-minimum displaced Kerr states. The necessary displacement parameter has been represented as a center of a crescent-shaped phase region and expressed in terms of the lowest moments of the ${\mathrm{\ensuremath{\Phi}}}_{\mathit{s}\mathit{c}\mathit{r}\mathit{A}}$ quasidistribution.