A FlyerTalk Posting Legend
Join Date: Sep 2009
Location: Minneapolis: DL DM charter 2.3MM
Programs: A3*Gold, SPG Plat, HyattDiamond, MarriottPP, LHW exAccess, ICI, Raffles Amb, NW PE MM, TWA Gold MM
Posts: 100,417
IIRC the SQ free stopover is for a short visit, intended for several days but a week might be the limit. A big part of the promo (at least for the past several years) has been the discounted hotel package, which presumably you won't want if you have family or a home there. Why not call SQ and ask? Or read the fare rules (for fares with these stopovers) and T&C for the offer carefully.
You seem to have a Singapore passport, but are you an official resident of the USA? If so, the stopover is more likely to apply to you. I'm assuming that you won't be a resident of Australia by then since you're planning to get the visa in Singapore.
Will an airline board you in the USA for a ticket USA-Singapore-Australia if you don't have a visa (or at least ETA if applicable for your passport) for Australia and the visa is needed for the second leg of your trip? Will not having a RT ticket or a ticket leaving Australia be an issue if that is a normal entry requirement?
Of course, if Singapore citizens can visit Australia without visas or ETAs, then many of these concerns disappear.