In New York, the main indexes initially slipped before regaining ground to end slightly higher.
In Europe, London's FTSE 100 and Frankfurt's DAX closed up slightly, with Paris's CAC 40 closing lower.
Oil prices gave up some of their recent gains, and the dollar remained under pressure from expectations of narrowing interest rate differentials.
US Federal Reserve Chairman Jerome Powell on Friday sent stock markets soaring back to near-record levels -- and the dollar plunging to its lowest in a year -- when he signaled a rate cut was coming.
Attention this week has turned to how large the cut could be, with investors seeing few reasons to push prices too aggressively.
The markets are seeking "more clarity from US economic data about the amount that the Fed may be prepared to ease in September and in the coming months," Jane Foley, head of foreign exchange strategy at Rabobank London, told AFP.
Analysts said trading was thin, partly out of caution and partly because of the coming Labor Day holiday weekend in the United States.
US second-quarter economic growth updates are due Thursday, followed by the Fed's preferred gauge of inflation on Friday and jobs data next week.
Investors will be looking for clues about the size of the promised rate reduction, which the Fed is expected to deliver at its next meeting on September 17-18.
Traders are also awaiting the latest earnings results from chip titan Nvidia on Wednesday, which could provide indications about demand for the hardware powering the AI boom.
"The company has a long track record of beating analysts' expectations but faces increasingly challenging comparisons," said Joe Mazzola, a strategist at Charles Schwab. "Results from Nvidia could help set the tone for the rest of the week."
In European business news, Spain's government blocked a takeover offer from a Hungarian consortium for Spanish train manufacturer Talgo.
Trading in Talgo shares was briefly suspended on the Madrid exchange but they still fell almost nine percent.
In Asia, Hong Kong rose despite losses in the tech sector that came after Temu owner PDD posted disappointing revenue figures and warned on the outlook for future growth.
The e-commerce firm's shares, which are listed in New York, were down 4.1 percent on Tuesday, extending losses after tanking Monday by a record 28.5 percent. This wiped tens of billions of dollars off its market capitalization
Also on the Hong Kong exchange, rivals Alibaba and JD.com both sank around four percent.
Oil had risen Monday after an exchange of fire Sunday between Iran-backed Lebanese militant group Hezbollah and Israel raised fears of a broader conflict. Prices have dropped back since.
The decline was a "likely technical correction" as price "held on to most of the recent gains driven by ongoing supply-side concerns amid a near-total production stoppage in Libya, along with escalating tensions between Israel and Hezbollah," Trading Economics said in a comment.
The eastern-based administration in oil-rich Libya said Monday that it would close fields under its control and suspend production and exports "until further notice" amid rising tensions with the UN-recognised government based in Tripoli.
New York - Dow: FLAT at 41,250.50 (close)
New York - S&P 500: UP 0.2 percent at 5,625.80 (close)
New York - Nasdaq Composite: UP 0.2 percent at 17,754.82 (close)
London - FTSE 100: UP 0.2 percent at 8,345.46 points (close)
Paris - CAC 40: DOWN 0.3 percent at 7,565.78 (close)
Frankfurt - DAX: UP 0.4 percent at 18,681.81 (close)
Tokyo - Nikkei 225: UP 0.5 percent at 38,288.62 (close)
Hong Kong - Hang Seng Index: UP 0.4 percent at 17,874.67 (close)
Shanghai - Composite: DOWN 0.2 percent at 2,848.73 (close)
Dollar/yen: DOWN at 143.96 yen from 144.53 yen on Friday
Euro/dollar: UP at $1.1185 from $1.1166
Pound/dollar: UP at $1.3261 from $1.3184
Euro/pound: DOWN at 84.34 pence from 84.64 pence
West Texas Intermediate: DOWN 2.4 percent at $75.53 per barrel
Brent North Sea Crude: DOWN 2.3 percent at $79.55 per barrel
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