Nearly six years after home prices started falling, more U.S. housing markets appear to be nearing a new phase: a prolonged bottom.

Hitting a bottom, of course, isn’t the same as a full-fledged recovery, which is still years off for many housing markets—as well as for millions of people who purchased homes or took cash out during the bubble.

The good news is that housing construction and home sales appear to have hit a floor. Home builders cut back heavily in the past four years and began construction on just 434,000 single-family homes last year, the lowest level on record. Research firm Zelman & Associates estimates builders will start construction on 540,000 homes this year, a 24% increase.

New-home sales during the first quarter posted double-digit gains from the previous-year period. A rebound here is likely simply because “we’ve starved the market for new-home construction,” said Ivy Zelman, the firm’s chief executive.

Sales of previously owned homes, meanwhile, are up 32% from their low point of mid-2010, when sales plunged following the expiration of home-buyer tax credits.

That leaves prices as the last measure that hasn’t yet stopped falling. But there are signs of progress on that front, too, as the pace of declines is slowing.

In February, home prices fell by 2% from their level of a year earlier, according to CoreLogic Inc., a real-estate-data firm. But after excluding foreclosures and other distressed sales, prices were down by just 0.8%, the smallest year-over-year decline since May 2010.

“If you remove the distress, you’re looking at housing prices not falling much further,” said Jonathan Miller, president of Miller Samuel Inc., a New York-based appraisal firm.

The problem, of course, is that foreclosures are still a very high share of sales in many of the hardest-hit markets.

One of the biggest headwinds today is the “shadow inventory” of potential foreclosures. Banks owned about 450,000 properties at the end of March, but there were an additional two million loans in some stage of foreclosure and around 1.7 million more where mortgage payments hadn’t been made in more than 90 days.

Housing economists are debating whether that shadow inventory will spoil any housing recovery. “That’ll be like a ball and chain,” said Mark Fleming, chief economist at CoreLogic. “It won’t prevent a recovery, but it could drag it out over several years.”

Housing is getting a lift from reduced supply and stronger demand. Mortgage-interest rates at near-record lows and prices at their 2002 levels have made homes more affordable than at any point in the past decade. The number of homes for sale has fallen over the past year. A top complaint of some real-estate agents today is that there aren’t enough homes to show potential buyers.

Ms. Zelman, who was among the first to warn that the air had begun seeping out of the housing bubble six years ago, said the shadow inventory is “not going to result in the double dip that people always talk about.” She points to a burgeoning appetite for housing from investors, who are scooping up homes that can be converted to rentals, and six years of pent-up demand from traditional buyers who feel better about their financial prospects. “The fear is gone,” she said.

While the foreclosure overhang is serious, some economists say there is a less-noticed tailwind that could balance things out: the sharp decline in new construction over the past four years. “A lot of the people who talk about ‘shadow inventory’ don’t talk about how slow the overall housing stock has been growing,” said Thomas Lawler, an independent housing economist in Leesburg, Va.

There’s more that will keep home prices from rising, once they do hit bottom. First, many Americans don’t have the required down payment or can’t qualify for a mortgage. Banks are making borrowers jump through more hoops in order to produce loans that can’t be subjected to a costly “buy-back” demand from Fannie Mae, Freddie Mac or other investors if the loan defaults. That is keeping credit very tight.

More than one-third of all homeowners have less than 25% equity, including 15% that are underwater, meaning their homes are worth less than what they owe.

Second, inventory declines may be less of a sign of health than they would suggest and instead reflect one of the structural problems holding back housing: Sellers are frozen, either unwilling or unable to sell at current values. Markets above the entry level, where demand from investors and first-time buyers isn’t as strong, face a particularly steep climb because of that equity hole.

“Nobody’s voluntarily putting their home up for sale,” said John Burns, a home-builder consultant based in Irvine, Calif.

For-sale inventories may also be lean because banks sharply decelerated the foreclosure process over the past 18 months after courts found that they were routinely passing off forged paperwork to take back homes. If prices are stabilizing because of that temporary drop in foreclosures, some recent gains will prove artificial. Ultimately, because that shadow inventory is concentrated in certain markets, price drops also will be concentrated there.

Markets that are more quickly absorbing the stock of foreclosures amid an improving economy, such as Phoenix, and in those where the overhang wasn’t as severe, such as Denver and Washington, D.C., have probably hit bottom. Others face a longer haul, particularly in overbuilt cities such as Atlanta and Las Vegas.

Many cities in Florida and the Northeast, where banks have been unable to satisfy court-administered foreclosure processes, have a glut of foreclosures that has yet to be digested.

A housing “recovery” has already had false starts. Two years ago, government stimulus gave a short-lived boost, and today, cheap mortgages and foreclosure delays could be doing the same.

There is plenty that can still go wrong—especially a sudden rise in mortgages rates or slowdown in job growth—and a lot that needs to go right for housing to recover. But there are more signs than there were a year ago that housing isn’t getting worse, and that it may slowly be getting better.