We study closures of conjugacy classes in the Lie algebras of the orthogonal and symplectic groups and determine which ones are normal varieties. Furthermore we give a complete classification of the minimal singularities which arise in this context, i.e. the singularities which occur in the open classes in the boundary of a given conjugacy class. In contrast to the results for the general linear group ([KP1], [KP2]) there are classes with non normal closure; they are branched in a class of codimension two and give rise to normal minimal singularities. The methods used are (classical) invariant theory and algebraic geometry.