NEW YORK — Even some of the cities that suffered the most in the housing bust are showing signs of improvement, with prices beginning to recover in places like Miami, Atlanta, and Detroit, according to the latest housing data.

In fact, there is improvement across the board, with home prices nationally inching up over their levels a year ago for the first time since 2010, when sales were fueled by a temporary tax credit for home buyers.

The report, known as the S&P/Case-Shiller Home Price Index, tracks prices nationally and in 20 major cities. Both showed gains over the past year, from June to June — 1.2 percent nationally, and 0.5 percent in the 20 cities.

Although prices were still depressed in a few of the cities, compared to a year earlier, every one of them showed price gains between May and June. The report is highly respected because it tracks actual price differences as a home is sold and resold over time.

Home prices are still down almost a third from their peak in 2006, but the bulk of the recent data reports are pointing to a slow recovery and increased optimism that could encourage potential buyers to take the plunge.

In Atlanta, the city with the biggest one-year decline in home prices, the market perked up by about 4 percent in May and again in June, according to nonseasonally adjusted numbers. Detroit saw price increases of 6 percent from May to June. A rise in prices is to be expected in June, a time when the market ordinarily heats up, but analysts called these increases particularly strong.

Housing went from being a huge engine of growth to a drag on the economy as inventories swelled, foreclosures mounted, and prices crashed. More recently, economists have pointed to a slow turnaround, saying it is once again contributing to the nation’s economic output.

A number of factors have helped: Investors have bought up properties on the cheap; reduced inventory has prompted competitive bidding; and lower interest rates have emboldened buyers — at least, those who can qualify for a mortgage.

The number of foreclosures has also declined as banks respond to tightened oversight in some jurisdictions.

But economists caution about potential dangers that could still hold home buyers back. Consumer confidence unexpectedly fell in August, largely because of diminished expectations for the future. The November election and the ‘‘fiscal cliff’’ of expiring tax breaks and looming spending cuts at the end of the year may also be keeping a damper on things.

Ed Stansfield, chief property economist for Capital Economics, warned the eurozone crisis could set off another financial disaster that would keep people from buying homes.

“Even so,’’ he wrote, ‘‘with prices rising at an annualized rate of close to 10 percent in the second quarter, without another major shock, house prices are more likely to surprise on the upside than the downside this year.’’

Some of the strongest showings could be seen in the Sun Belt, where many cities were among those hit hardest by the crash.

Places that showed relatively small declines, like Denver and Dallas, have shown correspondingly small upticks in prices. In Washington, which is somewhat shielded from recessions by the presence of the federal government, homes lost a third of their value but have recovered by more than 14 percent.