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I assume you are a US citizen working in Singapore, and will answer your question on that basis. Please let me know if that is not the case.
As a US citizen, you would be responsible for taxes in both countries, however, you would also be allowed some credits to offset those taxes to help avoid or eliminate the double taxation issue. This would be the case even if you had no family living in this country. A US citizen is always subject to US taxes on his worldwide income, regardless of where he lives.
First, if you actually reside outside of the US for at least 330 days in a 12 month consecutive period, then you are entitled to claim the Foreign Earned Income Exclusion. For the 2009 tax year that amount was $91,400. That means that your first $91,400 in foreign earnings is exempt from US taxation. The Foreign Earned Income Exclusion is claimed by filing Form 2555 with your US return.
If you do not meet the requirement to claim the Foreign Earned Income Exclusion, then even though the full amount of your foreign income would be subject to US tax, at the same time you would be allowed to claim a credit for the foreign taxes paid to Singapore on that same income. The foreign tax credit is claimed by filing Form 1116 with your US tax return.
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Thank you ncdc