* U.S. service sector grows, housing prices vault higher * Euro zone economy starting to recover, Germany leads way * European services firms most optimistic in eight months * Chinese services sector at 4-mnth peak, business confidence rises By Steven C. Johnson NEW YORK, Feb 5 (Reuters) - The vast U.S. services sector expanded again last month, extending a three-year run of growth, while European business optimism hit an eight-month high, suggesting the euro zone economy was starting to recover. Rising corporate confidence also helped China's services sector grow at its quickest pace in four months, providing more evidence of a gradual rebound in the world's second largest economy. The data on Tuesday bolstered the view that things were looking up for the world economy, a sentiment that has lifted stock markets around the globe and pushed the U.S. benchmark S&P 500 to five-year highs. "I think people are optimistic right now," said Michael Woolfolk, senior currency strategist at BNY Mellon in New York. "The view is you want to buy risky assets on good data and ignore the bad data." While the Institute for Supply Management's service sector index eased slightly in January, it still showed the sector continued to grow for a third straight year. A report from data analysis firm CoreLogic said U.S. home prices rose for a 10th straight month in December, with the year-on-year jump the biggest since May 2006 HOPE AND WORRY IN EUROPE The outlook in Europe was more subdued, though surveys on Tuesday offered evidence that the worst of the downturn in the 17-country euro zone may have passed. Markit's Eurozone Composite PMI, based on business activity across thousands of companies, and a good gauge of economic growth, rose in January to a 10-month high of 48.6 from 47.2 in December. While still below the 50 mark that divides growth and contraction, its rise in January was its third straight. Private industry makes up nearly two-thirds of the euro zone's economy. Germany, the euro zone's biggest economy, took the lead, with the PMI chalking up its biggest one-month rise since August 2009 to soar to its highest level since June of 2011. The reading for the bloc's second-biggest economy France, however, plummeted to its lowest in nearly four years, below readings from perennial laggards Spain and Italy. "That's a really worrying sign. It's going to cause more tensions between Germany and France ... on various aspects of euro zone management," said Jennifer McKeown, economist at consultancy Capital Economics. "France will continue to call for more supportive policy, not just for the periphery but the euro zone as a whole. Germany is taking a much more hard-line stance." The euro zone economy probably contracted 0.4 percent at the end of last year, notching up its third negative quarter, and is likely to stagnate in the current period, according to a Reuters poll published last month. BRITISH, CHINESE FIRMS SEE GROWTH Activity rose among Britain's services firms, which account for more than three-quarters of gross domestic product. "The UK services PMI has bounced back into growth territory, dampening fears over an unprecedented triple-dip recession being called," said James Knightley at ING. After only one quarter of expansion, Britain's economy contracted again at the end of last year. That put it on the brink of its third recession in four years. In China, the HSBC services purchasing manger's index rose to 54 last month, with service sector firms confident about the future. The business expectation index -- one of many accompanying sub-indices that do not contribute to the overall PMI business activity reading for the services sector -- rose to its highest since May 2012 at 64.4. "China's growth recovery is now on a firmer footing," said Qu Hongbin, a HSBC economist. "Solid job gains plus higher business expectations bode well for further improvement of the services sector's growth."