Singapore has a very low crime rate; it's safe to walk around unaccompanied at night, and people often leave their bags unattended. Bear in mind, however, that low crime doesn't mean no crime. Use common sense: be careful of your handbag in busy pedestrian areas and packed trains. Don't wear a money belt or a waist pack, both of which peg you as a tourist. Distribute your cash and any valuables (including your credit cards and passport) between a deep front pocket, an inside jacket or vest pocket, and a hidden money pouch. Do not reach for the money pouch once you're in public.
Jaywalk at your own risk; Singapore drivers won't slow down or stop to let you cross.