Where's a good place for me to invest using AUD in Singapore?

That is either one long stint or one well paying job!

Depends on the focus of your question - if you want to keep the AUD exposures it means Aussie market and there is plenty of choices equity wise (and managed funds for real estate).

ASX has a bunch of ETFs which you can easily get various types of exposures. So if you are looking outside of AU, as a mature market has the usual characters (real estate, major indicies). But in that case you might as well repatriate the cash back to SG and invest in local providers?​​​

There are lots of great companies listed on ASX!

Hotcopper is a good forum to explore! :)