  In this work, we investigate the fine regularity of L\'evy processes using
the 2-microlocal formalism. This framework allows us to refine the multifractal
spectrum determined by Jaffard and, in addition, study the oscillating
singularities of L\'evy processes. The fractal structure of the latter is
proved to be more complex than the classic multifractal spectrum and is
determined in the case of alpha-stable processes. As a consequence of these
fine results and the properties of the 2-microlocal frontier, we are also able
to completely characterise the multifractal nature of the linear fractional
stable motion (extension of fractional Brownian motion to {\alpha}-stable
measures) in the case of continuous and unbounded sample paths as well. The
regularity of its multifractional extension is also presented, indirectly
providing an example of a stochastic process with a non-homogeneous and random
multifractal spectrum.
