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Observations of the interstellar medium (ISM) show a complex density and velocity structure which is in part attributed to turbulence. We here present a self-contained introduction to the multifractal formalism in a microcanonical version which allows us for the first time to compute precise turbulence characteristic parameters from a single observational map without the need for averages in a grand ensemble of statistical observables. We focus on studying the 250 mu-m Herschel map of the Musca filament and make use of MHD simulations. We find a clear signature of a multiplicative cascade in Musca with an inertial range from 0.05 to 0.65 pc. We show that the proposed microcanonical approach provides singularity spectra which are truly scale invariant as required to validate any method to analyze multifractality. The obtained, for the first time precise enough, singularity spectrum of Musca is clearly not as symmetric as usually observed in log-normal behavior. We claim that the ISM towards Musca features more a log-Poisson shape of its singularity spectrum. Since log-Poisson behavior is claimed to exist when dissipation is stronger for rare events in turbulent flows in contrast to more homogeneous (in volume and time) dissipation events, we suggest that this deviation from log-normality could trace enhanced dissipation in rare events at small scales, which may explain or is at least consistent with the dominant filamentary structure in Musca. Moreover we find that sub-regions in Musca tends to show different multifractal properties. It strongly suggests that different types of dynamics exist inside the Musca cloud. These differences between sub-regions appear only after eliminating noise features which have the tendency to "log-normalize" an observational map. Our study sets up fundamental tools which will be applied to other galactic clouds and simulations in forthcoming studies.