I think you will find that once you get into semi serious competitors that they use different guns for every disapline. I would also guess that not many serious shooters shoot all 3 sports, it's hard to be at the top of any one game when you are trying to shoot 3 of them.
Myself I would not want to use my trap gun for skeet or sporting clays, it's too heavy, a little too long, and not ideal for quick mounting.
For sporting I'd want something lighter, probably with a little shorter length of pull and a parallel comb, that doesn't shoot quite as high, with slightly shorter barrels (say 30").
For skeet the same gun for sporting might do but you'd probalby want shorter barrels yet, maybe 26".
Of course you could have a custom setup made that would have say 4 barrel sets and 2 stocks and probably shoot all 3 just fine. Though the cost might be very high, though possilby not as high as 3 different guns.
Myself I put most of my effort into trap, so I have a gun that is suitable (but not ideal) for skeet or sporting when I shoot those games. However I don't take either of those that serious.