Given positive integers k <= m and a graph G, a family of lists L = {L(v) : v in V(G)} is said to be a random (k,m)-list-assignment if for every v in V(G) the list L(v) is a subset of {1, ..., m} of size k, chosen uniformly at random and independently of the choices of all other vertices. An n-vertex graph G is said to be a.a.s. (k,m)-colourable if lim P(G is L-colourable) = 1, where L is a random (k,m)-list-assignment. We prove that if m >> n^(1/k^2) D^(1/k) and m >= 3 k^2 D, where D is the maximum degree of G and k >= 3 is an integer, then G is a.a.s. (k,m)-colourable. This is not far from being best possible, forms a continuation of the so-called palette sparsification results, and proves in a strong sense a conjecture of Casselgren. Additionally, we consider this problem under the additional assumption that G is H-free for some graph H. For various graphs H, we estimate the smallest m0 for which any H-free n-vertex graph G is a.a.s. (k,m)-colourable for every m >= m0. This extends and improves several results of Casselgren. Based on joint work with Michael Krivelevich.