Consider any dense r-regular quasirandom bipartite graph H with parts of size n and fix a set of r colours. Let L be a random list assignment where each colour is available for each edge of H with probability p. We show that the threshold probability for H to have a proper L-edge-colouring is p of order (log n)/n. This answers a question of Kang, Kelly, Kühn, Methuku and Osthus. We thus obtain the same threshold for Steiner Triple Systems and Latin squares; the latter answers a question of Johanssen from 2006.