Three that I regularly operate from I use Vx unless Vy is the best choice for the conditions on the day. ie turbulence, windshear etc I don't use Vx in heavy 200 series cessna as they are not as forgiving as the 100 series when the engine stops. Piper Slab wings I always land ahead as the world can always do without one more Cheroclunk
1 Takeoff from one airfield that departs over a nasty piece of open sea - making it back to land or very close to shore is the priority, returning to the strip is not possible however a clear area is along the coastline. Height v Distance
2 Takeoff uphill toward gently rising terrain. Ground rising under aircraft.
3 Short (300 meter) bush surrounded strip with a safe crash landing at the strip or in a river valley behind and to the side of the strip.
They are probably extreme cases but are examples were a different thought process than simple "must do this" rule apply.
One other that I can think of would be an extra long strip where a straight ahead landing is possible. (not the best one I could think of but another reason Vx could be better than Vy in that situation)