The problem of constructing flexible stochastic models to describe the variability in shape of solid particles is challenging. Natural objects often exhibit mono- or multi-fractal features, i.e. irregular shapes and self-similar patterns. This paper presents a general framework for modelling three-dimensional star-shaped particles with a locally variable Hausdorff (or fractal) dimension. In our approach, the radial function of the particle is represented by an anisotropic Gaussian random field on the sphere. We additionally derive a simulation algorithm being parenthetical to the spectral turning bands method proposed in Euclidean spaces. We illustrate the use of our proposal through numerical examples, including a multifractal simulated version of the Earth topography.