Singapore has the fastest growing population among all the countries I checked on the Asian Development Bank Key Indicators for Asia and the Pacific 2010. You will find it here on the bank's website.
I didn't check Samoa and Vanuatu and a few other Pacific islands on the list. But, among all the other countries, Singapore posted the biggest population increase in 2008 and 2009, according to the figures given by the bank. It shows by what percentage the population increased in each country. Those are the figures used here in the chart and the following table, which includes all the countries I checked.