Paying for what class, and wanting to fly in what class?

Ie, if int'l upgrades to Asia are not a concern, I'd say AA, because of its strength in South America on its own metal. That covers GRU and EZE, and they also fly to ZRH. (All of these can be flown on one connection, with a choice of at least two connecting cities at present if flying all AA metal. AA has dropped SWISS as a partner, so I don't know if they'll continue flying two of their own planes to ZRH each day, but I would think one is likely to continue, plus you can always fly AA to LHR and then get a codeshare on BA from there.)

You have to use OneWorld alliance partners to SIN and HKG (which means you cannot upgrade there). OW partner Cathay Pacific (with AA codeshares available) will get you to HKG from LAX nonstop, and to SIN via HKG.

But no matter which ariline you choose, I think you'll have to use alliance partners to somewhere, as I don't think there's any US airline that flies its own metal to all those cities.

And AA has the advantage that all miles from all sources count toward lifetime status (1MM = lifetime Gold, 2MM = lifetime Plat). Even if you earn very few miles from non-flying, just the fact that all your bonus miles (including elite status bonuses) will count toward lifetime status means you can get to lifetime status much faster than on any other airline I know of (all other airlines that have lifetime status only count EQMs AFAIK).

If you do choose AA and you don't yet have status with them, look into the Challenge (which can get you elite status really quickly your first year on such long int'l flights). There's tons of info on this in a Sticky at the top of the AA forum on FlyerTalk.