Aims: We examine the interstellar medium (ISM) of M 33 to unveil fingerprints of self-gravitating gas clouds throughout the star-forming disk. Methods: The probability distribution functions (PDFs) for atomic, molecular, and total gas surface densities are determined at a resolution of about 50 pc over regions that share coherent morphological properties and considering cloud samples at different evolutionary stages in the star formation cycle. Results: Most of the total gas PDFs are well fit by log-normal functions whose width decreases radially outward. Because the HI velocity dispersion is approximately constant throughout the disk, the decrease in PDF width is consistent with a lower Mach number for the turbulent ISM at large galactocentric radii where a higher fraction of HI is in the warm phase. The atomic gas is found mostly at face-on column densities below NHlim = 2.5 × 1021 cm-2, with small radial variations of NHlim. The molecular gas PDFs do not show strong deviations from log-normal functions in the central region where molecular fractions are high. Here the high pressure and rate of star formation shapes the PDF as a log-normal function, dispersing self-gravitating complexes with intense feedback at all column densities that are spatially resolved. Power-law PDFs for the molecules are found near and above NHlim, in the southern spiral arm and in a continuous dense filament extending at larger galactocentric radii. In the filament nearly half of the molecular gas departs from a log-normal PDF, and power laws are also observed in pre-star-forming molecular complexes. The slope of the power law is between -1 and -2. This slope, combined with maps showing where the different parts of the power law PDFs come from, suggests a power-law stratification of the density within molecular cloud complexes, in agreement with the dominance of self-gravity.