  Recently, a new approach in the fine analysis of stochastic processes sample
paths has been developed to predict the evolution of the local regularity under
(pseudo-)differential operators. In this paper, we study the sample paths of
continuous martingales and stochastic integrals. We proved that the almost sure
2-microlocal frontier of a martingale can be obtained through the local
regularity of its quadratic variation. It allows to link the H\"older
regularity of a stochastic integral to the regularity of the integrand and
integrator processes. These results provide a methodology to predict the local
regularity of diffusions from the fine analysis of its coefficients. We
illustrate our work with examples of martingales with unusual complex
regularity behavior and square of Bessel processes.
