  Consider a hyperelliptic curve of genus $g$ over a field $K$ of
characteristic zero. After extending $K$ we can view it as a marked curve with
its $2g+2$ Weierstrass points. We present an explicit algorithm to compute the
stable reduction of this marked curve for a valuation of residue characteristic
$2$ over a finite extension of $K$. In the cases $g\le2$ we work out relatively
simple conditions for the structure of this reduction.
