The best way to find a place to rent in Singapore is to contact a housing agency. Housing agents have large databases of available property, and will be able to find you an apartment that best suits you needs based on location and price.
In many cases it can be cheaper to communicate with landlords directly. However, professional housing agents put your needs first and will help you with leasing procedures that may be unfamiliar to you. A list of accredited estate agents can be found on the website of the Singapore Estate Agencies Association (SEAA) http://www.seaa.org.sg/ . Online property classifieds can also be very helpful. Many websites will have postings for rooms and apartments available for rent, and in these cases you can contact the landlord directly. The following website is an example of an online portal, and has a section where you can search for rental apartments or condos in Singapore, http://www.singaporeexpats.com/. You can also look on Just Landed's service for property and rentals in Singapore.
Alternative accommodation, such as hotels and hostels, are also available. However, these tend to be better for short stays or until permanent accommodation is found as some hotels can cost up to $200 a night.