  In this paper we provide a local Cauchy theory both on the torus and in the
whole space for general Vicsek dynamics at the kinetic level. We consider
rather general interaction kernels, nonlinear viscosity and nonlinear friction.
Particularly, we include normalised kernels which display a singularity when
the flux of particles vanishes. Thus, in terms of the Cauchy theory for the
kinetic equation, we extend to more general interactions and complete the
program initiated in Gamba et. al. (2016) (where the authors assume that the
singularity does not take place) and in Figalli et. al. (2017) (where the
authors prove that the singularity does not happen in the space homogeneous
case). Moreover, we derive an explicit lower time of existence as well as a
global existence criterion that is applicable, among other cases, to obtain a
long time theory for non-renormalised kernels and for the original Vicsek
problem without any a priori assumptions. On the second part of the paper, we
also establish the mean-field limit in the large particle limit for an
approximated (regularized) system that coincides with the original one whenever
the flux does not vanish. Based on the results proved for the limit kinetic
equation, we prove that for short times, the probability that the dynamics of
this approximated particle system coincides with the original singular dynamics
tends to one in the many particle limit.
