* MSCI Asia ex-Japan -0.4% after hitting 3-month peak * Oil holds ground above $70; U.S. dollar pinned near lows * Focus on U.S. jobs data, upcoming Fed and ECB meetings SHANGHAI/SINGAPORE, June 2 (Reuters) - Stocks chopped around record peaks on Tuesday, while the dollar was pinned near recent lows, as markets awaited U.S. jobs data and looked ahead to crucial central bank meetings in Europe and United States for guidance on the interest rates outlook. MSCI’s broadest index of Asia-Pacific shares outside Japan touched a three-month peak, before profit-taking in recently strong Chinese markets pulled it 0.4% lower. Aussie stocks hit a record high, while Japan’s Nikkei rose 0.5%. World stocks sat just below Tuesday’s record, however momentum has clearly ebbed from the year-long COVID-19 rally as investors begin to worry a stronger-than-expected rebound means sooner-than-expected monetary policy tightening. Brent crude was firm after closing above $70 a barrel for the first time in two years, while bond and currency markets were becalmed as traders wait on data for clues as to the recovery’s progress. U.S. and European equity futures were steady, with S&P 500 futures flat and EuroSTOXX 50 futures up 0.2%. “Non-farm payrolls looms large on Friday,” said National Australia Bank’s head of currency strategy, Ray Attrill, and could rouse financial markets from a few weeks of inertia. After a big miss in April, when monthly hiring of 266,000 confounded expectations for 1 million, May estimates range between 400,000 and 1 million, with consensus around 664,000. “The uncertainty around payrolls will have an influence on sentiment...whether we get a print nearer to 250,000 or a million is going to have quite some impact I think.” European factory gate price surveys and the U.S. Federal Reserve’s “beige book” economic snapshot, both due later on Wednesday, will also be closely watched while Fed and European Central Bank meetings later in June loom largest on the horizon. “As the major developed economies continue to reopen from COVID lockdowns the focus on central bank meetings is going to intensify,” MUFG analysts said in a monthly outlook note. They expect the ECB to avoid signalling a slowdown in bond purchases, but think the Fed might confirm that “very initial” discussions on tapering its bond buying have begun. DATA WATCH Australia provided the latest piece of better-than-forecast economics on Wednesday, with growth beating market expectations and lifting national output above pre-pandemic levels. That helped the stock market to its latest record, but couldn’t kick the currency out of its recent range as the central bank has been stubbornly sticking to its dovish tone. The Aussie was last steady at $0.7747. The euro held at $1.2218, just shy of recent highs as the risk of a dollar rally if strong data prods the Fed into tightening is keeping investors from adding to large dollar shorts. Overnight, data showed U.S. manufacturing activity rose in May, as pent-up demand boosted orders, pushing Treasury yields higher. Benchmark 10-year rates were steady on Wednesday at 1.6079%, with the small rise enough to crimped gold prices. At $1,898 an ounce the metal was flat, but it is up more than 13% in just over two months, mostly thanks to help from a falling U.S dollar. Brent futures rose 0.6% to $70.67 per barrel and U.S. West Texas Intermediate crude added 0.56% to $68.10 per barrel, despite the OPEC+ alliance agreeing to hike output in July. Wall Street indexes had closed broadly steady on Tuesday, with the revival of gains in retail-investor driven “meme stocks” standing out. AMC Entertainment Holdings rose more than 20% and is up more than 1,400% for the year - perhaps helped by a collapse in cryptocurrencies redirecting small traders’ focus to the equity market. Bitcoin bought $36,750 on Wednesday. Reporting by Andrew Galbraith in Shanghai and Tom Westbrook in Singapore; Editing by Lincoln Feast. Our Standards: The Thomson Reuters Trust Principles.