What is the most effective way to transfer over $10,000 from the U.S. to Singapore while getting a better exchange rate?

You can transfer money from Singapore to US, via Wire Transfer, or even via Western Union/Money Gram (albeit the latter has a US$ 10,000 limit).

For wire transfer to the US, you need the following:

Beneficiary's Bank's Name
Address
Account Title
Account Number
ABA Routing #
SWIFT Code

If they are using a correspondent bank, then you would need,

Correspondent Bank's Name
Correspondent Bank's Address
Beneficiary Bank's Account Title in Correspondent Bank
Beneficiary Bank's Account Number in Correspondent Bank's
ABA Routing #
SWIFT Code

Your wire transfer company would know exactly how to route this. Although, it is not a bad idea to take the international wire transfer instructions from the bank.