  Let $A(n,m)$ be a graph chosen uniformly at random from the class of all
vertex-labelled outerplanar graphs with $n$ vertices and $m$ edges. We consider
$A(n,m)$ in the sparse regime when $m=n/2+s$ for $s=o(n)$. We show that with
high probability the giant component in $A(n,m)$ emerges at
$m=n/2+O\left(n^{2/3}\right)$ and determine the typical order of the 2-core. In
addition, we prove that if $s=\omega\left(n^{2/3}\right)$, with high
probability every edge in $A(n,m)$ belongs to at most one cycle.
