* Asian stock markets : https://tmsnrt.rs/2zpUAr4
* Factory activity up in China, Japan and S.Korea
* ECB seen easing on Thurs, Canada may cut on Wed
* Focus on US ISM surveys, May payrolls reports
* Oil prices see-saw as OPEC+ extends most output cuts
SYDNEY, June 3 (Reuters) - Asian share markets rallied on Monday, as investors looked forward to an interest rate cut in Europe and quite possibly Canada as the next step in global policy easing, though sticky inflation threatens to make the process a drawn out affair.
There was also better news from China as the private Caixin survey showed a pick-up in its main factory index to a two-year top of 51.7 in May, from 51.4 in April.
Japan's factory activity expanded for the first time in a year in May, while activity in South Korea grew at the fastest pace in two years.
All of which helped MSCI's broadest index of Asia-Pacific shares outside Japan bounce 1.4%, having slid 2.5% last week. Chinese blue chips added 0.3%.
Japan's Nikkei rose 1.1%, after rebounding from one-month lows on Friday, while South Korea gained 1.8%.
Meanwhile, EUROSTOXX 50 futures climbed 1.0% and FTSE futures 0.8%, as the risk-on mood spread.
South Korean President Yoon Suk Yeol flagged the possibility of a vast amount of oil and gas reserves in the sea off the country's east coast.
Indian markets hit record highs on wagers Prime Minister Narendra Modi will expand his alliance's majority in parliament when election results are released on Tuesday, leading to greater economic reforms.
Month-end flows saw Wall Street stage a late rally on Friday and left the Nasdaq up almost 7% for May. Early on Monday, S&P 500 futures were up 0.2%, with Nasdaq futures adding 0.3%.
The prospect of lower borrowing costs globally has been generally positive for equities.
ECB TO PIP FED
The European Central Bank (ECB) is considered almost certain to trim rates by a quarter point to 3.75% on Thursday, the first time in history it would have eased ahead of the U.S. Federal Reserve.
However, a surprisingly high reading for euro zone inflation out last week blunted hopes for a rapid round of reductions and markets have 57 basis points of easing priced in for this year.
"The probability of back-to-back cuts now appears very low, putting the focus for a second move on September," said Bruce Kasman, head of economic research at JPMorgan.
"We suspect President Christine Lagarde will signal that the direction of rates is downward next week, but the policy statement will emphasize that future moves are data-dependent, and there is no pre-commitment to a particular rate path."
Markets also imply around an 80% chance the Bank of Canada will cut at its meeting on Wednesday and 59 basis points of easing this year, though analysts are hopeful the easing will be even deeper.
Investors are a lot less dovish on the Fed, seeing little prospect of a move until September and even that is far from a done deal.
The outlook could change this week given data due includes key surveys on services and manufacturing, and the May payrolls report where unemployment is seen holding at 3.9% as 190,000 net new jobs are created.
In forex markets, the Japanese yen remains the weakest of the majors, though the government is clearly prepared to spend big to slow its slide. Data out last week showed Tokyo spent 9.788 trillion yen ($62.27 billion) on currency intervention between April 26 and May 29.
The dollar firmed to 157.41 yen, just short of last week's peak at 157.715. The euro held firm at $1.0855 , still benefiting from the EU inflation report, but faces resistance at $1.0895.
Gold was a shade softer at $2,322 an ounce, having now rallied for four months in a row helped in part by buying from central banks and China.
Oil prices see-sawed after OPEC+ agreed on Sunday to extend most of its oil output cuts into 2025, though some cuts will start to be unwound from October 2024 onwards.
Brent eased 10 cents to $81.01 a barrel, while U.S. crude lost 6 cents to $76.93 per barrel. ($1 = 157.1900 yen)
(Reporting by Wayne Cole; Editing by Jamie Freed, Christopher Cushing and Rashmi Aich)
