Found this on one of those question and answer sites which seems to deal with the problems you have been having with buying from Amazon. It does not make items ship here that Amazon wont ship, but it will help you identify items that will ship here.
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Amazon will ship to Australia but the experience can be a little confusing and the biggest problem is finding the items that Amazon will ship there. This often frustrates many and somepeople assume that Amazon wont ship at all or they wont ship a particular category.
Essentially you can ship dvd's, books, clothing, health, tools. watches, jewelry, shoes, electronics, handbags, toys and more to most countries outside the USA including Australia.
There is a way to make this process as easy as possible if you follow a few easy steps
*** Step 1
Visit the official Amazon Global page at
http://amzn.to/Amazon-Global-Shipping - This is not the main Amazon homepage so make sure you record the address so you can come back to this page whenever you want.
This is still not perfect as not everything listed here will be able to be shipped to Australia but everything listed can be shipped out of the USA to somewhere so we still have to do steps 2 to 5. Buy following this stem We have however removed millions of items that Amazon will never be able to ship outside the USA.
*** Step 2
Login to an Amazon Account before you start shopping. If you dont yet have an account create one.
*** Step 3
Go into your account settings, click on Manage Addresses and setup your address in Singapore and then set it as the 1-Click default address. Your address has to be setup before you start shopping as it queries this address to see if the item can be shipped to you. Without this you will not find out if an item cannot be shipped to you until you checkout.
*** Step 4
Go Shopping - as you look at every item it will check to see if it can be shipped to your address. If it can't it will indicate so in red writing.
*** Step 5
International shipping rates by many merchants can be excessive so it is best to only buy from Amazon who have low shipping rates. In the left hand column click Amazon in the seller box to make sure you are only seeing items sold and shipped by Amazon