Let H1 and H2 be graphs. A graph G has the constrained Ramsey property for (H1, H2) if every edge-colouring of G contains either a monochromatic copy of H1 or a rainbow copy of H2. Our main result gives a 0-statement for the constrained Ramsey property in G(n, p) whenever H1 = K1,k for some k ≥ 3 and H2 is not a forest. Along with previous work of Kohayakawa, Konstadinidis and Mota, this resolves the constrained Ramsey property for all non-trivial cases with the exception of H1 = K1,2, which is equivalent to the anti-Ramsey property for H2. For a fixed graph H, we say that G has the anti-Ramsey property for H if any proper edge-colouring of G contains a rainbow copy of H. We show that the 0-statement for the anti-Ramsey problem in G(n, p) can be reduced to a (necessary) colouring statement, and use this to find the threshold for the anti-Ramsey property for some particular families of graphs.