  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, K\"uhn, 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.
