You've probably heard that Singapore has the best food in Asia - whatever that's supposed to mean. One thing that's for sure, though, is that Singapore offers you a great variety of different foods that you cannot expect to find in most other Asian cities. Here are the places to go:
Hawker Centers offer you the cheapest food in Singapore - you will end up paying between S$2 and S$5 for a meal, unless you go to the hawkers that tourists like to go to. Let's start with those: Lau Pa Sat (located in the CBD) is the most beautiful hawker center in the city, because it was kept in its colonial setting. Lau Pa Sat is where the famous Sate street opens at night, with numerous Sate stands on the street beside the actual Hawker center. At Lau Pa Sat you can also find some of the best stingray in Singapore, although some believe that some stands at Newton Circus do an even better job...go find out for yourself. Newton food court is the other famous one among tourists. Both Lau Pa Sat and Newton are open 24 hours. As you get around Singapore, you will find that there are many many other hawker centers, and soon you'll find your own favorite ones! The space here certainly does not suffice to describe them all!
Seafood: supposedly the most famous seafood place in town is Dino's seafood at the East Coast Seafood Center. Try the black pepper and the chily crabs there! There are dozens of other good sefood places along the East coast. For Singapore's only white pepper crab you'll have to go to No Signboard Seafood at the Oasis next to the stadium. It tastes pretty much exactly like the black papper crab.
Boat Quai and Clarke Quai offer you a whole bunch of restaurants - if you don't mind having tourists all around you, and paying a slight premium for the location.
The cheapest Sushi in town (in a restaurant, and only according to my knowledge), can be bought at a Sushi place on Orchard Road between Dhoby Ghaut and Somerset MRT...but I forgot the name. Otherwise you can always go to one of Sakae Sushi's outlets, which offer you a great variety of reasonably priced Sushi.
Singapore is known for its buffets and all-you-can-eat events. The easiest way to find out about them is to check with the major hotels (who offer supreme quality food at prices that are usually in the S$30 to S$40 range).
For ethnic food you should check out Little India, the area around Arab Street and Geylang!