  In 1991, Bollob\'{a}s and Frieze showed that the threshold for $G_{n,p}$ to
contain a spanning maximal planar subgraph is very close to $p = n^{-1/3}$. In
this paper, we compute similar threshold ranges for $G_{n,p}$ to contain a
maximal bipartite planar subgraph and for $G_{n,p}$ to contain a maximal planar
subgraph of fixed girth $g$.
