  We obtain sufficient conditions for the emergence of spanning and
almost-spanning bounded-degree {\sl rainbow} trees in various host graphs,
having their edges coloured independently and uniformly at random, using a
predetermined palette. Our first result asserts that a uniform colouring of
$\mathbb{G}(n,\omega(1)/n)$, using a palette of size $n$, a.a.s. admits a
rainbow copy of any given bounded-degree tree on at most $(1-\varepsilon)n$
vertices, where $\varepsilon > 0$ is arbitrarily small yet fixed. This serves
as a rainbow variant of a classical result by Alon, Krivelevich, and Sudakov
pertaining to the embedding of bounded-degree almost-spanning prescribed trees
in $\mathbb{G}(n,C/n)$, where $C > 0$ is independent of $n$.
  Given an $n$-vertex graph $G$ with minimum degree at least $\delta n$, where
$\delta > 0$ is fixed, we use our aforementioned result in order to prove that
a uniform colouring of the randomly perturbed graph $G \cup
\mathbb{G}(n,\omega(1)/n)$, using $(1+\alpha)n$ colours, where $\alpha > 0$ is
arbitrarily small yet fixed, a.a.s. admits a rainbow copy of any given
bounded-degree {\sl spanning} tree. This can be viewed as a rainbow variant of
a result by Krivelevich, Kwan, and Sudakov who proved that $G \cup
\mathbb{G}(n,C/n)$, where $C > 0$ is independent of $n$, a.a.s. admits a copy
of any given bounded-degree spanning tree.
  Finally, and with $G$ as above, we prove that a uniform colouring of $G \cup
\mathbb{G}(n,\omega(n^{-2}))$ using $n-1$ colours a.a.s. admits a rainbow
spanning tree. Put another way, the trivial lower bound on the size of the
palette required for supporting a rainbow spanning tree is also sufficient,
essentially as soon as the random perturbation a.a.s. has edges.
