Sights in Singapore are covered in more detail under the various districts. Broadly speaking:
Beaches and tourist traps: Head to one of the three beaches on Sentosa. Other beaches include the East Coast and the Southern Islands (boat service from Harbourfront Shopping Centre).
Culture and cuisine: See Chinatown for Chinese treats, Little India for Indian flavors, Kampong Glam (Arab St) for a Malay/Arab experience or the East Coast for delicious seafood, including the famous chilli and black pepper crab.
History and museums: The area north of the Singapore River is Singapore's colonial core, with historical buildings and museums.
Nature and wildlife: The Singapore Zoo, Night Safari , Jurong Bird Park and Botanical Gardens are all in the North and West section. Other well known areas are the Bukit Timah Nature Walk which joins the MacRitchie Reservoir, Pulau Ubin and the Sungei Buloh Wetland Reserve.
Skyscrapers and shopping: The heaviest shopping mall concentration is in Orchard Road, while skyscrapers are clustered around the Singapore River, but also check out Bugis to see where Singaporeans shop.
What To Do
While you can find a place to practice nearly any sport in Singapore — golfing, surfing, scuba diving, even ice skating — due to the country's small size your options are rather limited and prices are relatively high. For watersports in particular, the busy shipping lanes and sheer population pressure mean that the sea around Singapore is murky, and most locals head up to Tioman (Malaysia) or Bintan (Indonesia) instead. See also Habitatnews and WildSingapore for news and updates about free tours and events.
Singapore has recently been experiencing a spa boom, and there is now plenty of choice for everything from holistic Ayurveda to green tea hydrotherapy. However, prices aren't as rock-bottom as in neighbors Indonesia and Thailand, and you'll generally be looking at upwards of $70 even for a plain one-hour massage. Good spas can be found in most five-star hotels and on Orchard, and Sentosa's Spa Botanica also has a good reputation. There are also numerous shops offering traditional Chinese massage, which are mostly legit, and "health centres", which are mostly not.
On the cultural side of things, Singapore has been trying loosen up and attract more artists and performances. The star in Singapore's cultural sky is the Esplanade theatre by the Riverside, a world-class facility for performing arts like classical music. Pop culture options are more limited and Singapore's home-grown arts scene remains rather moribund, but any bands and DJs touring Asia are pretty much guaranteed to perform in Singapore. Advance tickets for almost any cultural event can be purchased from SISTIC, either online or from any of their numerous ticketing outlets, including the Singapore Visitor Centre on Orchard Rd.