  The noncommutative space of light-like worldlines that is covariant under the
light-like (or null-plane) $\kappa$-deformation of the (3+1) Poincar\'e group
is fully constructed as the quantization of the corresponding Poisson
homogeneous space of null geodesics. This new noncommutative space of geodesics
is five-dimensional, and turns out to be defined as a quadratic algebra that
can be mapped to a non-central extension of the direct sum of two
Heisenberg-Weyl algebras whose noncommutative parameter is just the Planck
scale parameter $\kappa^{-1}$. Moreover, it is shown that the usual time-like
$\kappa$-deformation of the Poincar\'e group does not allow the construction of
the Poisson homogeneous space of light-like worldlines. Therefore, the most
natural choice in order to model the propagation of massless particles on a
quantum Minkowski spacetime seems to be provided by the light-like
$\kappa$-deformation.
