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.