  We study the local regularity and multifractal nature of the sample paths of
jump diffusion processes, which are solutions to a class of stochastic
differential equations with jumps. This article extends the recent work of
Barral {\it et al.} who constructed a pure jump monotone Markov process with
random multifractal spectrum. The class of processes studied here is much
larger and exhibits novel features on the extreme values of the spectrum. This
class includes Bass' stable-like processes and non-degenerate stable-driven
SDEs.
