* Euro pinned at $1.3600 awaiting outcome of ECB meeting
* ECB seen cutting all its rates, but much already priced in
* HSBC/Markit May services PMI drops to 50.7, from 51.4
By Wayne Cole
SYDNEY, June 5 (Reuters) - Asian share markets grudgingly gave ground on Thursday while the euro flatlined at $1.3600, hostage to great expectations that the European Central Bank will finally end months of dithering by easing policy further.
Not helping was a drop in HSBC/Markit's measure of the China service sector to 50.7 in May from April's 51.4, though that was still above the 50-point level that is supposed to separate growth from contraction.
The findings buck the trend seen in other China surveys in recent days which suggested the economy may be stabilising after a weak start to the year and raise the question of whether Beijing needs to do more to put a floor beneath growth.
Japan's Nikkei shed early gains to be up just a sliver at 15,082 points, so far failing to extend a two-week rally to test the April top of 15,164.
MSCI's broadest index of Asia-Pacific shares outside Japan dipped 0.1 percent, while South Korea's market lost 0.6 percent.
Wall Street had been just as hesitant with the Dow ending up 0.09 percent, while the S&P 500 gained 0.19 percent and the Nasdaq 0.41 percent.
The economic outlook was no clearer after a mixed bag of U.S. data that included disappointing results on the trade deficit and private employment, but better news on the service sector.
Still, it was notable that JPMorgan's measure of global industry output boasted a strong rise of 1.5 points in May, hinting at a broadening and strengthening in world growth.
In currency markets, the euro held a tight orbit around $1.3600 awaiting the outcome of the ECB policy meeting at 1145 GMT and President Mario Draghi's news conference at 1230 GMT.
Economists in a Reuters poll expected the ECB to cut its main refinancing rate by 15 basis points to 0.10 percent and the deposit rate to -0.10 percent from zero.
Imposing a negative deposit effectively charges banks for parking their excess money at the central bank -- a step that may, or may not, encourage them to lend the money instead.
The ECB is also thought likely to launch a refinancing operation aimed at funding smaller business across the EU, but to stop far short of the sort of quantitative easing embraced by the U.S., UK and Japan.
However, with so much already priced in to markets it could be difficult for policymakers to proffer a positive surprise.
"We continue to think that risks going into ECB are biased to the central bank underwhelming the market and are tactically positioned for a bounce in EUR/USD," cautioned JPMorgan strategist Niall O'Connor.
"The ECB's tone and message will be just as if not more important than the alphabet soup of policy announced -- does this represent a step change or a sea change for the ECB?" he added.
"We believe attitudes will change only gradually but we'll have to wait and see."
Yet, expectations of ECB action have helped drive German bond yields lower, giving the dollar a larger rate advantage. Indeed, U.S. two-year Treasuries are now offering a premium of 34 basis points over their German counterparts , the fattest margin in almost seven years.
A break below the recent trough at $1.3585 could see the euro gap to support in the $1.3475/$1.3500 area. The U.S. dollar index was steady at 80.658, while the currency kept most of its recent gains on the yen at 102.60.
Benchmark 10-year Treasury yields were at 2.593 percent after rising to levels not seen since mid-May.
In commodity markets, copper was subdued after suffering its biggest one-day fall since mid-April amid jitters about the impact on financing deals from a probe at a Chinese port.
Benchmark copper was changing hands at $6,783 a tonne, having shed 1.2 percent on Wednesday.
Gold was idling at $1,243.15 an ounce, still pinned near a recent four-month trough of $1,240.61.
Brent crude for July delivery was last down 16 cents at $108.24 a barrel. U.S. crude eased 29 cents to $102.35 per barrel. (Editing by Shri Navaratnam & Kim Coghill)