  We show that the threshold for the binomial random $3$-partite, $3$-uniform
hypergraph $G^{3}((n,n,n),p)$ to contain a Latin square is $\Theta(\log{n}/n)$.
We also prove analogous results for Steiner triple systems and proper list
edge-colorings of the complete (bipartite) graph with random lists. Our results
answer several related questions of Johansson, Luria-Simkin,
Casselgren-H\"aggkvist, Simkin, and Kang-Kelly-K\"uhn-Methuku-Osthus.
