A BBQ and tools is a must. They don't sell it here, at least I have not found any after asking around. This is assuming you are going to get a place with a balcony. The BBQs provided here at most
condos charge you about $10 each time and you need to buy your own coals etc. Yeah, no gas BBQ!!!
If you have kids, kids clothing is also good as the local variety are somewhat different unless you only want American brand names, which can be very expensive for daily wear; unlike Target and Kmart etc.
If your Mrs is size 12 and above, stock up on some summer clothes. You have very limited stores that stock those sizes. Also, bras are a must for her unless she likes heavily padded bras and small sizes.
Do not worry about bringing your bedlinen unless you are bringing your own bed as the Queen or King size beds here are shorter than what we are used to.
Wine is always good as they are expensive here and or the cheaper variety. But you would need to buy a wine fridge when you get here or they will all go bad in a matter of days due to the humidity. However be mindful of the taxes.
If you have young children, bring all their stuff from car seats, cots etc as they are very expensive here and mainly American brands.
Buy some new bathers as Seafolly brand (available here) are extremely expensive.
Leather lounge suite if you like them as they are very expensive here. All lounge furniture are also very low back if youl ike that style or if you prefer some support for your neck/head, bring your own.
All that being said, you can buy most things here in Singapore but at a very high price as almost everything is imported whether it is from its local neighbours, Australia, US or UK. Since you have the room in the container, why not bring it in.
If you get a chance to come and look around before moving here, take the opportunity to hit the shops and you will find out what is available and at what price. If you don''t then try and find an Asian friend in Melb who is a migrant themselves for some guidance.
I hope this has helped. We have just moved here from Melb too for 6 months and still take home a shopping list each time we go back.