We say that a graph G is anti-Ramsey for a graph H if any proper edge-colouring of G yields a rainbow copy of H, i.e., a copy of H whose edges all receive different colours. In this talk we will determine the threshold at which the binomial random graph becomes anti-Ramsey for any fixed graph H, given that H is sufficiently dense. This extends the previously known cases where H is a clique, a cycle, or close to a regular graph with sufficiently many vertices. Our proof employs a graph decomposition lemma in the style of the Nine Dragon Tree theorem that may be of independent interest.

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