There are several attractions worth seeing in Singapore, and they can all be done in a day fairly easily. One of the easiest ways to get around is to hire a car for the day. We hired a Maxi Cab for S$35 an hour. Maxi Cabs can hold up to 8 people, so you don't have to take two or more cabs if you're traveling in a small group. You can also take public transportation to just about everywhere in the city, but Mt. Faber and Sentosa Island are a little far to take the bus.
Merlion Park, located at the mouth of the Singapore river, is the most famous statute in Singapore. It's a statute with the head of a lion and the body of a fish.
Esplanade Theater on the Bay, a short walk from the Merlion, is the concert theater in Singapore. It's a very nice building and attached to a mall and a modern art gallery.
Featured on the Amazing Race, the Fountain of Wealth is located in Suntec City, a very large mall. The fountain touts itself as the largest fountain in the world, and indeed it is very large. Though the schedule may change, usually in the morning the fountain is off so that visitors can walk around the center touching the water in the middle for good luck. In the afternoon, at approximately 12pm, the full fountain starts and it is a wonderful sight to see. At night, the fountain has a light show that plays in the water.
Mount Faber is a peak fairly close to the city and provides excellent views from the top. It also is one of two gateways into Sentosa Island, where there are numerous tourist attractions, such as the aquarium and night zoo.
There are other sights in Singapore, such as a Hindu temple in Chinatown, a very large Buddhist temple located out of the city, built without nails according to classic Chinese building methods, and the Raffels hotel, which locals will tell you is the first hotel built in Singapore, but these are all things that depend on your taste, as they're mostly for those interested in either religion and/or architecture.