This paper is devoted to second-order necessary optimality conditions for the Mayer optimal control problem when the control set U is a closed subset of Rm. We show that, in the absence of endpoint constraints, if an optimal control (U) over bar( u) is singular and integrable, then for almost every t such that (U) over bar( t) is in the interior of U, both the Goh and a generalized Legendre-Clebsch condition hold true. Moreover, when the control set is a convex polytope, similar conditions are verified on the tangent subspace to U at (U) over bar( t) for almost all t's such that (U) over bar( t) lies on the boundary. U of U. The same conditions are valid also for U having a smooth boundary at every t where (U) over bar (T) is singular and locally Lipschitz and (U) over bar (t).. U. In the presence of a smooth endpoint constraint, these second-order necessary optimality conditions are satisfied whenever the Mayer problem is calm and the maximum principle is abnormal. If it is normal, then analogous results hold true on some smaller subspaces.