Try this website out for an estimate of how much it costs to have somthing shipped from US to Singapore. Take note that it is only an estimate, and shipping costs tends to vary from seller to seller.
www.usps.com
If you have a good ebay record, and you pay via paypal, sellers sometimes tend to make the exception, but you must email them in advance for their agreement.
As for import tax, it only applies to anything that has a declared equivalent value of SGD400, and it is called GST. That is 4% for nothing. On the other hand, if the shipper does not declare the value, or declares it at a value of less than SGD400 equivalent (exchange rate is up to customs, not you), you pay nothing.
Also, you have to take into account insurance. If you do not buy any, and the item is lost in transit or is damaged, that will be your problem, not the seller. As such, if you get your purchase insured, and the value comes up to more than SGD400, you will have to pay that 4% GST.