(Adds U.S. market open, byline, dateline; previous LONDON)
* MSCI index, Dow, S&P 500 set fresh intra-day highs
* European shares slip as Catalonia uncertainty unnerves investors
* Spain's main index set for biggest daily fall in over a year
* U.S. Treasury yields rise on strong service sector data
By Herbert Lash
NEW YORK, Oct 4 (Reuters) - An index of global stocks edged up to a fresh intra-day high on Wednesday, helped by signs of strong economic growth worldwide, while the dollar pared losses on data showing the U.S. service sector accelerated in September to its fastest in 12 years.
U.S. Treasury debt yields rose on a stronger-than-expected reading for the Institute for Supply Management's index of non-manufacturing activity, which boosted the likelihood the Federal Reserve will raise interest rates in December.
Oil prices were mixed on caution that rising U.S. crude output could scupper a crude rally that lasted for most of the third quarter.
MSCI's gauge of global stock performance set a fresh intra-day high, while on Wall Street the Dow and benchmark S&P 500 also notched new intra-day highs.
The potential for tax reform has been driving the recent rally on Wall Street as geo-political tensions have ebbed and economic data remains strong, said Michael Arone, chief investment strategist at State Street Global Advisors in Boston.
"If the earnings season comes in as expected, you continue to see progress made on tax reform and you see diminished concerns about the geopolitical situation, the market wants to move higher," he said.
MSCI's index of equity performance in 47 countries gained 0.03 percent while the pan-European FTSEurofirst 300 index of leading regional shares lost 0.18 percent.
The Dow Jones Industrial Average rose 16.86 points, or 0.07 percent, to 22,658.53. The S&P 500 lost 0.24 points, or 0.01 percent, to 2,534.34 and the Nasdaq Composite dropped 10.04 points, or 0.15 percent, to 6,521.67.
European shares fell as the impact of the crisis in Catalonia spread from Madrid and Spanish banks to the wider industry and euro zone region, particularly Italy.
Spain's IBEX was set for its worst day in 14 months with a 2.6 percent decline.
Spanish government borrowing costs rose to their highest since March, stretching the gap over German benchmarks to the widest in over five months after Catalonia's secessionist leader said the region will declare independence in "days."
Catalonia will move next week to declare independence from Spain, a regional government source said, as a violence-marred vote on Sunday threatens the country's foundations and has unnerved financial markets.
Spain's 10-year bond yield rose as much as 7 basis points to 1.795 percent in early trades, according to Reuters data, the highest since mid-March.
Data from around the world showed a robust global economy.
U.S. private employers added 135,000 jobs in September, topping economists' expectations by 10,000, even as Hurricane Harvey and Irma "significantly impacted smaller retailers," a report by a payrolls processor showed.
The increase for the ADP National Employment Report was the smallest since October 2016. August private payroll gains were revised down to 228,000 from the original 237,000 increase.
In Europe, business across the euro zone grew rapidly in September as firms struggled to keep up with demand, a survey showed, with October looking likely to be lively as well.
IHS Markit's final composite Purchasing Managers' Index for the euro zone bounced to 56.7 last month from August's 55.7, in line with an earlier flash estimate and comfortably above the 50-mark that separates growth from contraction.
Japan's services sector expanded in September at the slowest rate in 11 months as the pace of new orders eased, but a raft of other data suggest the economic recovery remains intact even as momentum may have ebbed slightly in the third quarter.
Brent was last at $55.95 per barrel, down 5 cents on the day. U.S. crude rose 6 cents to $50.48 a barrel.
The dollar index fell 0.13 percent, with the euro up 0.15 percent to $1.176. The Japanese yen strengthened 0.14 percent versus the greenback at 112.73 per dollar.
Benchmark 10-year notes fell 2/32 in price to push yields up to 2.3390 percent.
Reporting by Herbert Lash