  We give a complete classification of semistable rank two sheaves on
three-dimensional projective space with maximal third Chern character. This
implies an explicit description of their moduli spaces. As an open subset they
contain rank two reflexive sheaves with maximal number of singularities. These
spaces are irreducible, and apart from a single special case, they are also
smooth. This extends a result by Okonek and Spindler to all missing cases and
gives a new proof of their result. The key technical ingredient is variation of
stability in the derived category.
