It is widely believed that both the stress and the sstress criteria for metric multidimensional scaling are plagued by the existence of nonglobal minimizers. At present, there is little theory that enlightens this belief. Trosset and Mathar (1997) established that nonglobal minimizers of the stress criterion can exist, while Glunt, Hayden, and Liu (1991) demonstrated that the distance matrices of all configurations for which the gradient of the sstress criterion vanishes lie on a certain sphere. This report extends existing theory in several directions. Emphasis is placed on the more tractable case of the sstress criterion. Because the stress and stress criteria must be minimized by numerical optimization, one result that is of immediate practical value is a simple device for improving the quality of the initial configurations from which optimization commences.