A colouring of a graph G = (V, E) is a function c : V → {1, 2,. . .} such that c(u) = c(v) for every uv ∈ E. A k-regular list assignment of G is a function L with domain V such that for every u ∈ V , L(u) is a subset of {1, 2,. .. } of size k. A colouring c of G respects a k-regular list assignment L of G if c(u) ∈ L(u) for every u ∈ V. A graph G is k-choosable if for every k-regular list assignment L of G, there exists a colouring of G that respects L. We may also ask if for a given k-regular list assignment L of a given graph G, there exists a colouring of G that respects L. This yields the k-Regular List Colouring problem. For k ∈ {3, 4} we determine a family of classes G of planar graphs, such that either k-Regular List Colouring is NP-complete for instances (G, L) with G ∈ G, or every G ∈ G is k-choosable. By using known examples of non-3-choosable and non-4-choosable graphs, this enables us to classify the complexity of k-Regular List Colouring restricted to planar graphs, planar bipartite graphs, planar triangle-free graphs and to planar graphs with no 4-cycles and no 5-cycles. We also classify the complexity of k-Regular List Colouring and a number of related colouring problems for graphs with bounded maximum degree.