if your bags are not checked through, and you need to collect and re-check, then you will need to clear customs and immigration, get bags and re-check them.
If your bags are checked through, then you just need to get between terminals on the little train thingy. Easy.
Not sure if Qantas will check your bags on SQ from SYD, but see if they do. If they do, they may not be able to issue you a SQ boarding pass at SYD. If that happens, when you get to SIN, just go to the transit counter in T3 and present your passport & bag check and they will print it for you. I did this in January when I went from coughet to Melbourne via SIN (Silkair onto Qantas) and it was very easy.