We construct static codimension-two branes in any odd dimension D, with a negative cosmological constant, and show that they are exact solutions of Chern-Simons (super) gravity theory for (super) AdS(D) coupled to external sources. The stability of these solutions is analyzed by counting the number of preserved supersymmetries. It is shown that static massive (D - 3)-branes are unstable unless some suitable gauge fields are added and the brane is extremal. In particular, in three dimensions, a 0-brane is recognized as the negative mass counterpart of the Banados-Teitelboim-Zanelli black hole. For these 0-branes, we write explicitly magnetically charged Bogomol'nyi-Prasad-Sommerfield states with various numbers of preserved supersymmetries within the OSp(p vertical bar 2) X OSp(q vertical bar 2) supergroups. In five dimensions, we prove that stable 2-branes with magnetic charge always exist for the generic supergroup SU(2, 4 vertical bar N), where N not equal 4. For the special case N = 4, in which Chern-Simons supergravity requires the addition of a nontrivial gauge field configuration in order to preserve the maximal number of degrees of freedom, we show for two different static 2-branes that they are Bogomol'nyi-Prasad-Sommerfield states (one of which is the ground state), and from the corresponding algebra of charges we show that the energy is bounded from below. In higher dimensions, our results admit a straightforward generalization, although there are presumably more solutions corresponding to different intersections of the elementary objects.