"My home is underwater, worth about $216,000 against a loan balance of $240,000. I can refinance the loan under the HARP program, reducing the rate from 5.75 percent to 5 percent, at a cost of $3,400. Should I refinance? Or just try to sell the home and bite the bullet?"
Your first step should be to clarify your options, which are more numerous than you indicate. There are at least two legitimate ways to give up the house, and if you stay put there are two ways to refinance.
Option 1 is to sell the house and pay the lender the difference between sale proceeds and loan balance. The advantage is that you are out clean, with no black mark on your credit record and no lingering debt. The disadvantage is that it will cost you $30,000-$40,000