Multifractal scaling (MFS) refers to structures that can be described as a collection of interwoven fractal subsets that exhibit power-law spatial-scaling behavior with a range of scaling exponents (concentration, or singularity strengths) and dimensions. The existence of MFS implies an underlying multiplicative (or hierarchical or cascade) process. Panoramic column density images of several nearby star-forming cloud complexes, constructed from IRAS data, are shown to exhibit such multifractal scaling, which we interpret as indirect but quantitative evidence for a nested hierarchical structure. The relation between the dimensions of the subsets and their concentration strengths (the "multifractal spectrum") appears to satisfactorily order the observed regions in terms of the mixture of geometries present, from strong pointlike concentrations, to linelike filaments or fronts, to space-filling, diffuse structures. This multifractal spectrum is a global property of the regions studied and does not rely on any operational definition of "clouds." The range of forms of the multifractal spectrum among the regions studied implies that the column density structures do not form a universality class, in contrast to indications for velocity and passive scalar fields in incompressible turbulence, providing another indication that the physics of highly compressible interstellar gas dynamics differs fundamentally from incompressible turbulence. There is no correlation between the geometrical properties of the regions studied and their level of internal star formation activity, a result that is also apparent from visual inspection. We discuss the viability of the multifractal spectrum as a measure of the structural "complexity" of the regions studied and emphasize the problematic dependence of all structural descriptors on the subjective preselection of the region to be described. A comparison of IRAS 100 μm column density (not intensity) images with ¹³CO, visual extinction, and C¹⁸O data suggests that structural details are captured by IRAS up to at least 30 mag of visual extinction, except in the vicinity of embedded stars, and that a lower column density connective structure not seen by other methods is revealed.