Unsurprisingly, it's already divided into three different aircraft - F-35A, F-35B, F-35C. They were supposed to be 70% common, but that's simply not achievable so the differences are quite a bit larger.
But for the obvious reasons they vastly prefer two engined planes, plus I've read the F-35's single engine is so big their current logistics system will have trouble getting new ones to a fleet at see, I assume this means it can't fit in a C-2 or maybe V-22.