The first thing you ought to know about
Cycling in Singapore is that the people here don't do it to commute - if they cycle at all, it is a form of recreation.
I have to point out that there is hardly any suitable road within downtown Singapore that you can cycle. You cannot bring your bicycle on board the MRT trains either. Throw in the blazing sun and swift traffic, and you have a recipe for disaster to attempt it in the Central Business District.
Having said that, there are places where you can cycle - for fun, that's it. Among these are the park areas along Changi Coast Road and East Coast Parkway. In addition, there are off-road paths at the Bukit Timah Nature Reserve where you can go bike riding. However, you probably would need a car anyway, to transport your bicycle there.