This paper reports the application of cellular automata to study the dynamic responses of Lamb-type problems for a tangential point load and a concentrated moment applied on the free surface of a half-plane. The medium is homogeneous, isotropic and linear elastic while having a random mass density field with fractal and Hurst characteristics. Both Cauchy and Dagum random field models are used to capture these effects. First, the cellular automata approach is tested on progressively finer meshes to verify the code against the continuum elastodynamic solution in a homogeneous continuum. Then, the sensitivity of wave propagation on random fields is assessed for a wide range of fractal and Hurst parameters. Overall, the mean response amplitude is lowered by the mass density field's randomness, while the Hurst parameter (especially, for β < 0.2) is found to have a stronger influence than the fractal dimension on the response. The resulting Rayleigh wave is modified more than the pressure wave for the same random field parameters. Additionally, comparisons with previously studied Lamb-type problems under normal in-plane and anti-plane loadings are given. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.