Let $G_{n,p}^{[κ]}$ denote the space of $n$-vertex edge coloured graphs, where each edge occurs independently with probability $p$. The colour of each existing edge is chosen independently and uniformly at random from the set $[κ]$. We consider the threshold for the existence of rainbow colored copies of a spanning subgraph $H$. We provide lower bounds on $p$ and $κ$ sufficient to prove the existence of such copies w.h.p.