The radiated energies, E_S, and seismic moments, M₀, for 942 globally distributed earthquakes that occurred between 1987 to 1998 are examined to find the earthquakes with the highest apparent stresses (τ_a=μE_S/M₀, where μ is the modulus of rigidity). The globally averaged τ_a for shallow earthquakes in all tectonic environments and seismic regions is 0.3 MPa. However, the subset of 49 earthquakes with the highest apparent stresses (τ_a greater than about 5.0 MPa) is dominated almost exclusively by strike-slip earthquakes that occur in oceanic environments. These earthquakes are all located in the depth range 7–29 km in the upper mantle of the young oceanic lithosphere. Many of these events occur near plate-boundary triple junctions where there appear to be high rates of intraplate deformation. Indeed, the small rapidly deforming Gorda Plate accounts for 10 of the 49 high-τ_a events. The depth distribution of τ_a, which shows peak values somewhat greater than 25 MPa in the depth range 20–25 km, suggests that upper bounds on this parameter are a result of the strength of the oceanic lithosphere. A recently proposed envelope for apparent stress, derived by taking 6 per cent of the strength inferred from laboratory experiments for young (less than 30 Ma) deforming oceanic lithosphere, agrees well with the upper-bound envelope of apparent stresses over the depth range 5–30 km. The corresponding depth-dependent shear strength for young oceanic lithosphere attains a peak value of about 575 MPa at a depth of 21 km and then diminishes rapidly as the depth increases. In addition to their high apparent stresses, which suggest that the strength of the young oceanic lithosphere is highest in the depth range 10–30 km, our set of high-τ_a earthquakes show other features that constrain the nature of the forces that cause interplate motion. First, our set of events is divided roughly equally between intraplate and transform faulting with similar depth distributions of τ_a for the two types. Secondly, many of the intraplate events have focal mechanisms with the T-axes that are normal to the nearest ridge crest or subduction zone and P-axes that are normal to the proximate transform fault. These observations suggest that forces associated with the reorganization of plate boundaries play an important role in causing high-τ_a earthquakes inside oceanic plates. Extant transform boundaries may be misaligned with current plate motion. To accommodate current plate motion, the pre-existing plate boundaries would have to be subjected to large horizontal transform push forces. A notable example of this is the triple junction near which the second large aftershock of the 1992 April Cape Mendocino, California, sequence occurred. Alternatively, subduction zone resistance may be enhanced by the collision of a buoyant lithosphere, a process that also markedly increases the horizontal stress. A notable example of this is the Aleutian Trench near which large events occurred in the Gulf of Alaska in late 1987 and the 1998 March Balleny Sea M= 8.2 earthquake within the Antarctic Plate.