We consider qualitative and quantitative properties of ``snapshot attractors'' of random maps. By a random map we mean that the parameters that occur in the map vary randomly from iteration to iteration according to some probability distribution. By a ``snapshot attractor'' we mean the measure resulting from many iterations of a cloud of initial conditions viewed at a single instant (i.e., iteration). In this paper we investigate the multifractal properties of these snapshot attractors. In particular, we use the Lyapunov number partition function method to calculate the spectra of generalized dimensions and of scaling indices for these attractors; special attention is devoted to the numerical implementation of the method and the evaluation of statistical errors due to the finite number of sample orbits. This work was motivated by problems in the convection of particles by chaotic fluid flows.