Tyler Cowen has a great post about hospital competition in Singapore and its virtues, which he argues is only partially attributable to health care policy per se. After all, one reason Singapore can have robust competition among hospitals is that it's densely populated (so most people live close to multiple hospitals) and also uses congestion pricing on roads so you can travel relatively easily. Still, he concedes that "not every hospital is within convenient reach in a country where many people do not have cars."
I would go a bit further in my praise of Singapore's policies in this regard by noting that one reason many Singaporeans get along OK without cars is that in addition to a metro and buses, Singapore makes sure to keep taxis cheap and widely available. In Singapore you pay $3 (Singapore dollars) for your first kilometer, then $0.22 per 400 meters for kilometers 2–10, then $0.22 per 350 meters beyond 10 kilometers. In New York, by contrast, the flag drop is $2.50 (in U.S. dollars) and then you pay $0.50 for every fifth of a mile—with a fifth of a mile being about 322 meters. Right now a Sinagapore dollar is worth 80 cents, so a Singapore cab trip starts out slightly cheaper than an NYC cab trip and the savings grow the longer your trip.
Now of course any city with regulated taxi fares (i.e., most cities around the world) could make fares cheaper quite easily by changing the regulations. But the risk is that with cheaper fares you end up with shortages—fares need to be high enough that people want to buy cars and either drive them or else hire drivers. But in most major cities, and certainly in New York, shortages of cabs are driven by regulations, not by market conditions. A taxi medallion in New York City costs more than $700,000 because the Taxi and Limosine Commission has made them so scarce. The city could easily lower fares and increase the availability of cabs simultaneously by issuing more medallions.
One reason that doesn't happen is the power of the taxi lobby. But another reason it doesn't happen is that American political discourse tends not to take cabs seriously as a mode of transportation. They're seen as something for tourists or a frivolous expenditure of the rich. But the case of Sinagpore and the hospitals is a key counterexample. Even in a city with good mass transit, people are sometimes going to want access to a car. Getting across town to the right hospital at a nonpeak hour while you're feeling ill is going to be a huge hassle even if your basic commuting and shopping needs are met by the mass transit system. To the extent that people are able to rely on plentiful, affordable taxicabs for those kind of "every once in a while" scenarios that arise, it becomes more viable to rely on transit for everyday transportation applications. Which is all just to say that Singapore is not only set up for you to be able to ride the bus to work, but also so that you won't have to ride the bus if you break your foot and need to go to the hospital. In a paradigm that doesn't rely on universal car ownership (Singapore, in fact, has very high taxes on car ownership to discourage it), you need a full spectrum of alternative transportation modes, including cars for hire.