FYI, it's perfectly legal at least on the U.S. side to mail (postally) U.S. currency. You usually do so at your own risk, although sometimes you can insure cash sent through postal mail. If you have a U.S. bank or U.S. credit union account in the United States, check with your bank first but usually you can mail them the cash to make a deposit, as long as the cash is a reasonable quantity -- US$1,300 would qualify.
Moderately large and larger quantities of cash and negotiable instruments must be declared when they cross borders. Both Singapore and the U.S. have that requirement.
Please cut down on your use of and retention of cash. It's nasty stuff, really, and it's getting nastier every day. Use low cost debit and credit cards, paid in full. ICBC Singapore issues a very good credit card for overseas (and even local) use, the Global Travel Mastercard, that will always beat cash in safety and value wherever accepted because it'll net you a ~0.5% rebate overseas.