US: Uber / Lyft
South East Asia: Grab / GoJek
Australia: Mainly just Uber (lots of smaller players but none with very significant market share)
If VCs keep funding Uber, Uber is unstoppable. But as soon as that money runs out, they're just another taxi company. Except they're a taxi company who loses $350 million dollars every month. That's not hard to compete against.
(I think UBER is comically over-valued, but I also think their core business is probably quite profitable.)
Being a more profitable taxi company doesn't mean much when most of your employees are also employees of your competitors, and most of your customers are also customers of your competitors. There's just no lock-in.
Considering they sold their business to Grab, I think you could safely say Uber+Grab are as close to a global service as you can get, and probably represents a significant fraction (though maybe not 50%+) of the global market