* Dollar steady after 2009 low vs euro, currency basket
* Global stock gains show improved risk appetite
* Data from U.S., China and Europe bolsters optimism
* RBA says scope for further easing remains, denting Aussie
By Rika Otsuka
TOKYO, June 2 (Reuters) - The dollar steadied on Tuesday after hitting its lowest this year against the euro and a basket of currencies the previous day as reassuring economic data and a rally in global shares reduced demand for the reserve currency.
The Australian dollar briefly slipped after the Reserve Bank of Australia held interest rates at a record low of 3 percent on Tuesday, but said there was scope to ease further if needed.
But traders said Australian dollar selling was light and was limited to profit-taking after the currency hit an eight-month peak against the U.S. dollar on Monday.
With commodity prices continuing to rise, the upward trend in commodity-linked currencies such as the Aussie remains intact, they added.
"A slide in the dollar in the past months first started as investors felt less need to seek the safety of the U.S. currency thanks to a rise in shares," said Hideki Hayashi, global economist at Mizuho Securities.
"In addition to that, the dollar is still vulnerable as investors are now taking more risks, shifting their money to the currencies of emerging countries and resource-producers."
The dollar index was steady at 79.147 after hitting its lowest since mid-December at 78.586 the previous day.
The euro was little changed from late U.S. trade at $1.4164, having rallied to $1.4246 on trading platform EBS on Monday, its strongest since late December.
The dollar fell 0.3 percent to 96.33 yen, after rising as high as 96.81 yen the previous day.
Tokyo's Nikkei share average hit an eight-month intraday high earlier in the day, following U.S. factory data which reinforced hopes that demand will stabilise.
The U.S. manufacturing sector improved in May to post its highest reading since September and construction spending in April notched its biggest increase in eight months, bolstering the view that the global economy is on the mend..
Generally upbeat manufacturing reports from the euro zone, Britain and China added to that optimism and fuelled a sell-off in the dollar on Monday.
As the RBA's rate decision has passed, the market is now eyeing policy announcements from the European Central Bank and the Bank of England expected on Thursday. Both are expected to hold short-term borrowing costs steady, and the ECB may provide more details on its plan to purchase assets to stimulate the economy.
DOLLAR UNDER PRESSURE?
Financial markets took in stride General Motors Corp's bankruptcy filing on Monday, the third-largest in U.S. history and the largest ever in the U.S. manufacturing sector.
"Still, as a general trend, players will likely opt for selling the dollar going forward," said Daisuke Uno, chief strategist at Sumitomo Mitsui Banking Corp.
"A string of factors that could prompt dollar selling still exist, including the U.S. credit problems and the restructuring of GM among others."
Higher risk currencies dipped from multi-month highs against the dollar as traders said their recent gains had been rapid and temporary profit-taking may be in store.
The Australian dollar was steady at $0.8102 after falling slightly following the central bank's rate announcement.
On Monday, the Aussie struck an eight-month high of $0.8154.
Sterling was little moved at $1.6430, off a seven-month high of $1.6497 hit on Monday. (Additional reporting by Aiko Hayashi; Editing by Joseph Radford)