Those green shoots that optimists keep seeing may be growing in the real estate market -- there are some indicators of healthier, more stable activity in housing.

In May, new housing starts jumped 17 percent over the previous month, and there was a 4 percent increase in building permits, too, according to new reports from the Commerce Department.

Housing inventories -- the backlog of homes on the market -- now stand at close to nine months. That is still high by historical standards, but lower than the 11-month levels reached at the end of 2007, says ZipRealty.com, a real estate listing and research firm.

But those inventories don't include the flood of foreclosures likely to hit the market soon. And with mortgage interest rates moving up (and they are), it's premature to believe that housing is poised to bounce back in a hurry.

Nonetheless, it's a good time for would-be buyers -- first-time homebuyers in particular -- to start checking out the market. Here's how to proceed.

-- Research your local market. Those national figures are almost meaningless when it comes to real estate. Markets in some parts of the country remain in free fall; others are already seeing bidding wars from speculators moving in to snap up underpriced homes. Talk to local real estate agents to discover how many months of inventory are on the market where you want to buy. Do research on websites like Trulia (www.trulia.com) that list pricing history and sales prices for many local markets. Discover where in the cycle your market is and how much fluff remains in prices. If inventories are falling and prices are rising, that is a solid sign that your neighborhood has bottomed out.

-- Take advantage of the first-time homebuyer tax credit. The Obama Administration's recovery package includes an $8,000 fully-refundable tax credit for first-time homebuyers who complete their purchases by December 1, 2009. To qualify as a first-time homebuyer, you can't have owned a home for three years, but that means some previous homeowners may qualify. For the full credit, your income must be less than $75,000 ($150,000 for couples filing jointly); it phases down for the next $20,000 of income. The Federal Housing Administration and some other local organizations are offering bridge loans to homebuyers so they can use these credits to pay for a downpayment or closing costs, and then pay back the bridge loan when they receive their annual tax refund in 2010.

-- Look for extra local incentives. California is offering $10,000 to people who buy newly built homes between now and March 2010. Washington, D.C., offers a $5,000 tax credit for first-time homebuyers moving into the city. Other states and municipalities may have other programs; your local real estate agent should know about them.

-- Buy, but remember not to stretch too much. Interest rates still are closer to their historic bottoms than their highs, so a fixed-rate mortgage will make sense for most buyers. If you go out 30 years, your payment, including principal, interest, property taxes and insurance, should not exceed a third of your monthly gross income. Before you buy the house, find out how much it costs to run: What are property taxes like? How much do the current homeowners spend on heating and air-conditioning?

-- Lowball your offer. Those houses with the most price resistance tend to be the cheapest ones in the market. But if you're buying a bigger or more expensive home -- especially one that has been on the market for months, or even years -- don't be afraid to submit a low, low offer. Do not worry about being insulting to the homeowners; you can say something nice like We love your beautiful home, but this is all we can afford. Most sellers are not rejecting low offers out of hand, but are countering with alternative offers. That gives you room to negotiate, or space to walk away and find another home closer to your price range.

-- Shop carefully for a mortgage and when you find a rate that you like and lock it in. If rates fall precipitously after you've locked in yours (unlikely), you can renegotiate or find another lender. But rates may continue to rise with economic recovery. To get a good rate on a mortgage, compare quotes from a local mortgage broker or bank, your own credit union, and a couple of online mortgage brokers. You can also compare mortgage rates at the sites of research firms like HSH Associates (www.hsh.com) and BankRate (www.bankrate.com). Make sure you are comparing not just rates but the long-term bottom-line cost of the mortgages; some will include points (extra fees that are paid up-front to secure a mortgage).

-- Control your closing costs too. It's a shame to work so hard researching and negotiating the home price and the loan only to give away thousands of extra dollars in myriad mystery closing costs. Don't just take the lender's word for it. At the Closing.com website (www.closing.com), you can see how to save money on items like the title search, title insurance and home appraisals. Make your lender give you a good-faith estimate of total closing costs, and don't be afraid to negotiate nuisance fees that you believe are unfair. The housing recovery, if that's what this is, is still fledgling and those sellers, agents and lenders really need your business more than you need theirs.

Linda Stern is a freelance writer. Any opinions in the column are solely those of Ms. Stern. You can e-mail her at lindastern@aol.com

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