In the city, taxis and buses are used most often, whereas when traveling for longer distances the SMRT (Singapore Mass Rapid Transit) is used. The system is modern and quite reliable, with few technical problems or strikes. A Tourist Pass can also be purchased for under S$10 a day, which permits unlimited use of the public transportation system. For permanent residents you can get a season pass called (see the TransitLink site below). Lastly, there is an airport shuttle that takes visitors from the airport directly to where they are staying. This service charges based on the amount of passengers.
Singapore's Changi Airport is a central traffic hub for Asia, and an important contributor to Singapore's economy. It is one of the best airports in the world, and is conveniently located with many transportation options to reach the inner city. Taxis, shuttles, busses, trains, and rental cars all can get you to and from the airport. There are also parking garages that make it easy for car owners or renters to store their cars.
More information: http://www.changiairport.com