We consider irreducible complex plane projective curves of degree six with simple singular points only and classify such curves up to equisingular deformation. (We concentrate on the so-called non-special curves, as the special ones are already known). We list all sets of singularities realized by such curves, discuss their relation to the maximizing sets (i.e., those of total Milnor number 19), and, for each set of singularities found, describe the connected components of the moduli space. We also discuss the question of the realizability of a given set of singularities by a real curve.