* Asian share markets mostly higher, bonds well supported
* Dollar on defensive after run of mixed US economic news
* Busy week for global indicators, highlighted by US payrolls
By Wayne Cole
SYDNEY, June 30 (Reuters) - Asian share markets edged cautiously higher on Monday while the dollar stayed under pressure ahead of packed week of economic data that will test investor hopes for a pick up in the U.S., and global, economy.
MSCI's broadest index of Asia-Pacific shares outside Japan was steady in morning trade. It gained 0.9 percent last week to be just under multi-month highs, aided by the prospect that interest rates in the major economies will remain near zero for many months to come.
The Tokyo market rose 0.35 percent, while South Korea added 0.55 percent.
Bulls are hoping to see evidence of an economic rebound in the United States in this week's busy calendar of data that includes the June payrolls report on Thursday, a day early due to the July 4 holiday.
Economists polled by Reuters on average expect jobs to rise 213,000 in June for the fifth straight month of gains above 200,000, a run unmatched since the Sept 1999-Jan 2000 period.
The U.S. earnings season also starts in the next couple of weeks, which will provide evidence on how the economy and profits are faring.
The Dow ended Friday 0.03 percent higher, though it was down 0.6 percent on the week. The S&P 500 gained 0.19 percent for the day, while the Nasdaq advanced 0.43 percent.
The FTSEurofirst300 suffered its first weekly loss in 10 weeks, while the MSCI world stocks gauge eased 0.3 percent for the week.
With stocks stalled, investors poured more money into bonds and had to lend for longer to earn any type of real return. That flow pushed bond prices up and yields down, particularly at the longer end of the curve.
Yields on 10-year Treasury notes were at 2.536 percent on Monday, having fallen almost 9 basis points last week. Likewise, yields on German 10-year paper dropped around 8 basis points to near record lows at 1.26 percent.
Bonds should remain supported given central banks in the U.S., Europe and Japan are seemingly committed to super low rates for the foreseeable future.
Federal Reserve Chair Janet Yellen gives a speech on "Financial Stability" on Wednesday and will take questions.
The European Central Bank holds its policy meeting on Thursday, a month after unleashing a far-reaching package of measures aimed at keeping the euro zone economy from slipping into a Japan-style deflation.
Consumer price figures for June are due later on Monday and should show inflation stuck at a low 0.5 percent, the ninth consecutive month in the ECB's "danger zone" of below 1 percent.
Globally, purchasing managers' indices (PMIs) for manufacturing are out on Tuesday and services on Thursday. They are expected to show a picture of growth or at least stability despite geopolitical tensions around Ukraine and Iraq.
In currencies, the dollar index was last at 80.042, after dipping as low as 80.010, a level not seen since May 21.
The euro was at $1.3645, near a 2-1/2 week high of $1.3652 set on Wednesday, while the dollar hovered just above a 5-week trough of 101.31 yen plumbed on Friday.
In commodity markets, gold was steady at $1,314.99 an ounce underpinned by geopolitical unrest in Iraq and Ukraine and the softer dollar.
Brent crude oil was down 16 cents at 113.14 a barrel, while U.S. crude futures lost 24 cents to $105.50. Oil prices have come off recent highs as fighting in Iraq stayed away from the country's south, where most of its oil is produced. (Editing by Shri Navaratnam)