Several European carriers launched test flights to challenge warnings that the volcanic ash cloud spreading across Europe from Iceland's Eyjafjoell volcano since Wednesday would destroy jet engines. Air France and KLM reported no problems.
A grouping of the continent's 36 major carriers called on governments for an "immediate reassessment" of the restrictions, saying they were having a "devastating impact" on the industry, and questioning whether they were proportionate.
"Airlines must be able to fly where it is safe to fly and make decisions accordingly. It is what our passengers demand of us," the Association of European Airlines said.
Meanwhile, KLM declared European airspace "safe" after a series of test flights, as the Dutch traffic authority allowed KLM to fly two cargo flights to Asia and the Middle East.
Air France said their first test flight, an Airbus A320 flying from Paris to Toulouse, "took place in normal conditions. No anomalies were reported."
However, French Ecology Minister Jean-Louis Borloo said the test flights were not steps towards reopening European airspace."We do tests and everything goes back to normal? No. It's not a theory which exists."
Two of Europe's three biggest airports, London Heathrow and Paris-Charles de Gaulle, remained shut.
But German authorities authorised the other, Frankfurt, plus five other national airports, to resume a limited service for several hours on Sunday, depending on the flight's direction.
Airports that had been closed in northern and eastern Spain, and airports in southern France, also reopened.
"Forecasts are that half of the flights planned (in Europe) could take place tomorrow," Spanish Secretary of State for European Affairs Diego Lopez Garrido told a press conference Sunday at the Eurocontrol flight coordinator.
However Britain later extended its flight ban until 1800 GMT on Monday and France ordered most airports shut until Tuesday morning.
About 30 countries have closed or restricted their airspace due to passenger safety fears, snagging 6.8 million passengers in a global backlog, according to the international airports council, ACI.
"More than 6.8 million passengers have been affected so far and European airports have lost close to 136 million euros (183 million dollars)," said its European head Olivier Jankovec, adding that a total of 313 airports had been paralysed by the restrictions.
More than 7,000 transit passengers have been stranded in the Gulf states of Dubai and Abu Dhabi, airlines said. Dubai-based Emirates Airlines said it was "providing accommodation and three meals a day for more than 5,000... transit passengers at a cost of more than one million dollars per day."
In Singapore, Changi airport and airlines have set aside special areas for stranded passengers to eat and sleep and are trying to provide them with free meals, sleeping bags and even tours of the city.
The enormous shroud of fine mineral dust particles now stretches from the Arctic Circle in the north to the French Mediterranean coast in the south, and from Spain into Russia.
The cloud is now heading toward Greece and into Russia, weather experts said.
"This is one of the most serious disruptions we have seen to air travel and our first priority must be the safety of passengers," British Prime Minister Gordon Brown told BBC television.
"The most important issue is making sure that it is safe to fly."
Some governments have set up emergency cells to work out how to get tens of thousands of stranded nationals home.
Around 63,000 flights have been cancelled in Europe since Thursday, the continent's air traffic coordinating agency Eurocontrol said. Some 20,OOO flights were cancelled on Sunday alone, according to a midday count.
Iceland's Eyjafjoell volcano started erupted on Wednesday, sending ash drifting towards Europe.
European Union transport ministers are to hold videoconference talks on the volcanic cloud on Monday with some officials expecting demands for special aid for airlines.
EU Commission chief Jose Manuel Barroso said he had ordered a full study to assess the impact of the situation created by the volcanic ash cloud on the economy, and the air travel industry in particular.
The closures stopped world leaders, including US President Barack Obama, from flying to the Polish city of Krakow for the state funeral of president Lech Kaczynski and his wife Maria.