Clark Quay, Boat Quay, Mohammed Sultan Rd - those are some of the bar areas. Just bring a lot of cash 'cause drinks are VERY pricey. Look for happy hours and ladies nights.
If you want big clubs, Ministry of Sound and Attica are two of the bigger clubs in Clark Quay. MoS is, as you would probably expect, is the biggest club in Singapore now. It just opened last year.
As for regular pubs and bars, all three areas also have those in abundance. Make sure you inquire as to their door policy. Some places have a high cover charge, but very cheap drinks. Other places have a high cover charge but it includes a couple of 'free' drinks. Some places have no cover but high drink prices. Enjoy, Singapore is a fun place - don't forget to try some Laksa from a street vendor or one of the Hawker centers.