  We construct explicit geometric models for and compute the fundamental groups
of all plane sextics with simple singularities only and with at least one type
$\bold E_8$ singular point. In particular, we discover four new sextics with
nonabelian fundamental groups; two of them are irreducible. The groups of the
two irreducible sextics found are finite. The principal tool used is the
reduction to trigonal curves and Grothendieck's {\it dessins d'enfants}
