  Given a number field $K$ and a finite set $S$ of places of $K$, the first
main result of this paper shows that the quadratic rational maps $\phi:{\mathbb
P}^1\to{\mathbb P}^1$ defined over $K$ which have good reduction at all places
outside $S$ comprise a Zariski-dense subset of the moduli space ${\mathcal
M}_2$ parametrizing all isomorphism classes of quadratic rational maps. We then
consider quadratic rational maps with double unramified fixed-point structure,
and our second main result establishes a geometric Shafarevich-type
non-Zariski-density result for the set of such maps with good reduction outside
$S$. We also prove a variation of this result for quadratic rational maps with
unramified 2-cycle structure.
