The dynamics of the deformations of a moving contact line is studied assuming two different dissipation mechanisms. It is shown that the characteristic relaxation time for a deformation of wavelength 2pi/|k| of a contact line moving with velocity v is given as tau(-1)(k)=c(v)|k|. The velocity dependence of c(v) is shown to depend drastically on the dissipation mechanism: we find c(v)=c(v=0)-2v for the case in which the dynamics is governed by microscopic jumps of single molecules at the tip (...