Observations of the interstellar medium (ISM) show a complex density and velocity structure, which is in part attributed to turbulence. Consequently, the multifractal formalism should be applied to observation maps of the ISM in order to characterize its turbulent and multiplicative cascade properties. However, the multifractal formalism, even in its more advanced and recent canonical versions, requires a large number of realizations of the system, which usually cannot be obtained in astronomy. We 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 (e.g., a temporal sequence of images). We compute the singularity exponents and the singularity spectrum for both observations and magnetohydrodynamic simulations, which include key parameters to describe turbulence in the ISM. For the observations we focus on the 250 µm Herschel map of the Musca filament. Scaling properties are investigated using spatial 2D structure functions, and we apply a two-point log-correlation magnitude analysis over various lines of the spatial observation, which is known to be directly related to the existence of a multiplicative cascade under precise conditions. It reveals 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 that are truly scale invariant, as required to validate any method used to analyze multifractality. The obtained singularity spectrum of Musca, which is sufficiently precise for the first time, is clearly not as symmetric as usually observed in log-normal behavior. We claim that the singularity spectrum of the ISM toward Musca features a more log-Poisson shape. 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 subregions in Musca tend to show different multifractal properties: While a few regions can be described by a log-normal model, other regions have singularity spectra better fitted by a log-Poisson model. This strongly suggests that different types of dynamics exist inside the Musca cloud. We note that this deviation from log-normality and these differences between subregions appear only after reducing noise features, using a sparse edge-aware algorithm, which have the tendency to "log-normalize" an observational map. Implications for the star formation process are discussed. Our study establishes fundamental tools that will be applied to other galactic clouds and simulations in forthcoming studies.