Expats in Singapore can definitely relax when it comes to matters of crime and safety: criminal activity, particularly of the violent kind, is very rare, and you don't have to worry about being in the wrong part of town at the wrong time of day. A noteworthy achievement for a global metropolis!
One of the most famous and widely known factoids about the island state is that crime is very rare. It is true that Singapore has one of the lowest crime rates not only in the Southeast Asian region, but worldwide. There are, of course, also some rumors floating around as to why that is the case. Contrary to what you have heard, you will not face prison time for spitting on the street, but punishments for even rather harmless crimes do tend to be rather severe in Singapore – a country which has not abolished corporal and capital punishment.