  We describe the multifractal nature of random weak Gibbs measures on some
class of attractors associated with $C^1$ random dynamics semi-conjugate to a
random subshift of finite type. This includes the validity of the multifractal
formalism, the calculation of Hausdorff and packing dimensions of the so-called
level sets of divergent points, and a $0$-$\infty$ law for the Hausdorff and
packing measures of the level sets of the local dimension.
