  Noncommutative spaces of geodesics provide an alternative way of introducing
noncommutative relativistic kinematics endowed with quantum group symmetry. In
this paper we present explicitly the seven noncommutative spaces of time-,
space- and light-like geodesics that can be constructed from the time-, space-
and light- versions of the $\kappa$-Poincar\'e quantum symmetry in (3+1)
dimensions. Remarkably enough, only for the light-like (or null-plane)
$\kappa$-Poincar\'e deformation the three types of noncommutative spaces of
geodesics can be constructed, while for the time-like and space-like
deformations both the quantum time-like and space-like geodesics can be
defined, but not the light-like one. This obstruction comes from the constraint
imposed by the coisotropy condition for the corresponding deformation with
respect to the isotropy subalgebra associated to the given space of geodesics,
since all these quantum spaces are constructed as quantizations of the
corresponding classical coisotropic Poisson homogeneous spaces. The known
quantum space of geodesics on the light cone is given by a five-dimensional
homogeneous quadratic algebra, and the six nocommutative spaces of time-like
and space-like geodesics are explicitly obtained as six-dimensional nonlinear
algebras. Five out of these six spaces are here presented for the first time,
and Darboux generators for all of them are found, thus showing that the quantum
deformation parameter $\kappa^{-1}$ plays exactly the same algebraic role on
quantum geodesics as the Planck constant $\hbar$ plays in the usual phase space
description of quantum mechanics.
