We examine the scaling properties of one-dimensional random walks on media with multifractal diffusivities, which is a simple model for transport in scaling porous media. We find both theoretically and numerically that the anomalous scaling exponent of the walk is dw=2+K(-1) where K(-1) is the scaling exponent of the reciprocal spatially averaged (dressed) resistance to diffusion. Since K(-1)>0, the walk is subdiffusive; the walkers are effectively trapped in a hierarchy of barriers. The trapping is dominated by contributions from a specific order of singularity associated with a phase transition between anomalous and normal diffusion. We discuss the implications for transport in porous media. ¿ 1998 American Geophysical Union