Compatibility summary
You will need a travel adapter in Singapore; plug types used there differ from Australia.
Plug comparison
Australia plug types: I
Singapore plug types: G
There is no common plug type between Australia (I) and Singapore (G). You need a travel adapter.
Voltage comparison
Australia: 230V, 50Hz
Singapore: 230V, 50Hz
Voltage difference is within a range many devices tolerate; check your appliance label.