What is the area with the largest velocity vertical shear in the ocean?
Areas associated with velocity shear at a stratified interface are notably strong in the region of the Equatorial Undercurrent, straits (ex: strait of Gibraltar, the Bosphorus) and regions of the thermocline where internal waves are strong (South China Sea).
These areas are typically associated with strong turbulence as a direct result of the strong velocity shear and shear instabilities that trigger Kelvin–Helmholtz billows (if the Richardson number is less than 1/4 for a sufficient amount of time) and subsequent hydrostatically unstable configurations and mixing.
Examples of magnitude of shear:

Maximum in Equatorial Undercurrent: $du/dz \approx 0.02$ s$^{-1}$
At the base of the mixed layer but above the core of the Equatorial Undercurrent the shear is the largest. See plot from Peters et al (1988) https://i.imgur.com/YbmAtV2.png

East of the Carmarinal Sill in the strait of Gibraltar: $du/dz \approx 0.024$ s$^{-1}$
In the Mediterranian, water flows westward into the Gulf of Cadiz as an undercurrent through the strait of Gibraltar, as these waters make a sharp turn northward, they may give rise to sub-surface eddies or &quot;Meddies&quot; when encountering canyons and underwater bathymetric features off the west coast of Portugal.

Very large amplitude internal waves in the South China Sea: $du/dz \approx 0.05$ s$^{-1}$
In the South China Sea very large amplitude internal solitary waves are ubiquitous features propagating on the thermocline with inherently large shear across the interface. See plot from Lien et al (2014)  https://i.imgur.com/qhQBI9g.png



References:
Ren-Chieh Lien, Frank Henyey, Barry Ma, and Yiing Jang Yang, 2014: Large-Amplitude Internal Solitary Waves Observed in the Northern South China Sea: Properties and Energetics. J. Phys. Oceanogr., 44, 1095–1115.
Fernando, Harindra JS. &quot;Turbulent mixing in stratified fluids.&quot; Annual Review of Fluid Mechanics 23.1 (1991): 455-493.
Thorpe, Steve A. The turbulent ocean. Cambridge University Press, 2005.
Peters, H., M. C. Gregg, and J. M. Toole. &quot;On the parameterization of equatorial turbulence.&quot; Journal of Geophysical Research: Oceans (1978–2012) 93.C2 (1988): 1199-1218.