  The celebrated canonical Ramsey theorem of Erd\H{o}s and Rado implies that
for a given graph $H$, if $n$ is sufficiently large then any colouring of the
edges of $K_n$ gives rise to copies of $H$ that exhibit certain colour
patterns, namely monochromatic, rainbow or lexicographic. We are interested in
sparse random versions of this result and the threshold at which the random
graph $G(n,p)$ inherits the canonical Ramsey properties of $K_n$. Our main
result here pins down this threshold when we focus on colourings that are
constrained by some prefixed lists. This result is applied in an accompanying
work of the authors on the threshold for the canonical Ramsey property (with no
list constraints) in the case that $H$ is an even cycle.
