* MSCI all-country world stocks index at highest since May 1
* Euro hovers near 2-week highs, yen weakens against dollar
* Coronavirus vaccine trial success sparks rally
* Oil steadies near 2-month high
* Graphic: World FX rates in 2020 tmsnrt.rs/2egbfVh
By Marc Jones
LONDON, May 19 (Reuters) - The euro and Italian government bonds continued on Tuesday to cheer German- and French-led plans for a 500 billion euro EU coronavirus recovery fund, though stock markets were suffering fatigue after their best day in months.
There was still a sense of optimism after Monday's news that early-stage tests on a possible COVID-19 vaccine had also proved encouraging, but the momentum had shifted.
Europe's STOXX 600 index gave up an early rise to slip 0.7% after surging 4% in the previous session, oil began to tread water and safe-haven U.S. government bonds were making ground again in the debt markets.
"The Franco-German proposals are ambitious, targeted and, of course, welcome," European Central Bank President Christine Lagarde said of Monday's plan, which would move the EU in the direction of a so-called 'transfer union'.
The euro was buying $1.0950, up more than 1% against the dollar since the plan was announced. It was also up near a two-month high against the Swiss franc, and options markets showed fewer traders were now betting against it.
After a sizeable drop in Italian borrowing costs, Spanish and Portuguese yields led the charge on Tuesday. Morgan Stanley's economists called the Franco-German proposal a "powerful common response, helping to mitigate the risk of a southern slump."
The Spanish 10-year yield fell 9 basis points to 0.715%, the lowest since early April, Portuguese yields hit their lowest since late March at 0.78% and Italy's briefly dipped under 1.6% at one point.
"It was a meaningful breakthrough but it is not going to be plain sailing from here," said Vasileios Gkionakis, Global Head of FX Strategy at Lombard Odier, cautioning that a number of northern EU countries had voiced resistance to the proposal.
Germany's monthly ZEW survey showed investor sentiment rebounding more quickly than expected though there were separate warnings that the German economy will slump over 7% this year.
Britain's pound shrugged off the UK's highest unemployment claims figures in nearly a quarter of a century, and a near 80% plunge in European new car sales in April contributed to 1.8% fall in auto sector shares.
Wall Street's S&P 500 futures were also pointing to modest early falls in New York after its strong rally.
A 10% rise in Walmart's first-quarter sales seemed to help the mood, though most traders were waiting on a Senate Banking Committee hearing later with Federal Reserve Chair Jerome Powell and U.S. Treasury Secretary Steven Mnuchin.
Asia's overnight moves had seen MSCI's broadest index of Asia-Pacific shares outside Japan jump 1.8% to two-week highs and Japan's Nikkei had added nearly 2% as the region followed Wall Street and Europe rallies.
Data from the first COVID-19 vaccine to be tested in the United States had shown it produced protective antibodies in a small group of healthy volunteers.
In the commodity markets, profit-taking saw Brent prune gains though the rally looked broadly intact amid signs that producers will stick to plans to cut output when global demand picks up.
Brent stood at $35.10 a barrel, more than double where it was in mid April, and U.S. crude was at $32.70 a month on from its collapse into negative territory.
"A powerful cocktail made of bullish ingredients have been supporting the oil market for a month ... Demand is improving, supply is decreasing," said oil broker PVM's Tamas Varga.
Additional reporting by Swati Pandey in Sydney and Noah Browning in London, Editing by Timothy Heritage