Context. The ubiquitous presence of filamentary structures in the interstellar medium asks for an unbiased characterization of their properties including a stability analysis. Aims. We propose a novel technique to measure the spectrum of filaments in any two-dimensional data set. By comparing the power in isotropic and anisotropic structures we can measure the relative importance of spherical and cylindrical collapse modes. Methods. Using anisotropic wavelets we can quantify and distinguish local and global anisotropies and measure the size distribution of filaments. The wavelet analysis does not require any assumptions on the alignment or shape of filaments in the maps, but directly measures their typical spatial dimensions. In a rigorous test program, we calibrate the scale dependence of the method and test the angular and spatial sensitivity. We apply the method to molecular line maps from magneto-hydrodynamic (MHD) simulations and observed column-density maps from Herschel observations. Results. When applying the anisotropic wavelet analysis to the MHD data, we find that the observed filament sizes depend on the combination of magnetic-field-dominated density–velocity correlations and radiative transfer effects. This can be exploited by observing tracers with different optical depth to measure the transition from a globally ordered large-scale structure to small-scale filaments with entangled field lines. The unbiased view to Herschel column-density maps does not confirm a universal characteristic filament width. The map of the Polaris Flare shows an almost scale-free filamentary spectrum up to the size of the dominating filament of about 0.4 pc. For the Aquila molecular cloud the range of filament widths is limited to 0.05–0.2 pc. The filaments in Polaris show no preferential direction in contrast to the global alignment that we trace in Aquila. Conclusions. By comparing the power in isotropic and anisotropic structures we can measure the relative importance of spherical and cylindrical collapse modes and their spatial distribution.