About 20 countries closed their airspace and some have extended flight bans into Monday. Experts say the cloud may not shift until later in the week.
Disruption is now said to be greater than that after 9/11 and the eruption shows no sign of abating.
Some airlines are making test flights to try to identify safe flight paths.
Test flights
Dutch airline KLM said it flew a Boeing 737-800 up to the usual maximum altitude of 13km (8 miles) on Saturday and Germany's Lufthansa said it flew 10 planes to Frankfurt from Munich at altitudes of up to 8km.
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COUNTRIES AFFECTED
Airspace closed:
Austria, Belgium, Bulgaria, Czech Republic, Denmark, Finland, France, Germany, Hungary, Ireland, Latvia, Luxembourg, Netherlands, Poland, Slovakia, Sweden, Switzerland, UK
Partial closures:
Italy (northern airspace closed until Monday)
Norway (limited flights in north)
Spain (northern airports closed)
Flights operating:
Greece, Portugal, Russia, Turkey, Ukraine
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KLM chief executive Peter Hartman, who was on board the plane, said there was "nothing unusual" about the flight.
"If the technical examination confirms this... we then hope to get permission as soon as possible to partially restart our operations," he added.
The tests were conducted at the request of the European Union to see whether travel disruption could be alleviated.
More trials are due to take place later on Sunday in countries including Britain, France and Italy.
UK Transport Secretary Lord Adonis, said "urgent discussions" were taking place between European and international agencies to ease the chaos.
"We want to be able to resume flights as soon as possible, but safety remains my paramount concern," he said.
Brian Flynn, head of operations at Eurocontrol, said aviation authorities were dealing with an "unknown phenomenon", but dismissed suggestions they were being over-cautious.
"Any risk of an aircraft penetrating an area that could have volcanic ash in it could have extreme safety consequences. And with the over-riding objective of protecting the travelling public, these exceptional measures have to be taken."
Weather experts say wind patterns mean the cloud is not likely to move far until later in the week.
The impact is likely to exceed the airspace shutdown after the 11 September 2001 attacks, the International Civil Aviation Organisation said.
Day turns to night as the dense ash cloud leaves Eyjafjallajokull in the dark
Airline losses
The International Air Transport Association (Iata) predicted little or no improvement on Sunday.
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Eurocontrol, which co-ordinates air traffic control in 38 nations, said some 17,000 flights were cancelled across Europe on Saturday, from a total of 22,000 on a normal day.
All but 55 of 337 scheduled flights by US carriers to and from Europe were also cancelled.
Airlines are estimated to be losing some £130m ($200m) a day in an unprecedented shutdown of commercial air travel.
BBC business editor Robert Peston said the disruption risked becoming a "major business and economic disaster" with several European airlines already facing financial difficulties.
"If [the disruption] goes on many days longer, a number of European airlines will run into financial difficulties and may need bailing out by governments - or so I am told by senior airline figures," our correspondent said.
European Commission chief Jose Manuel Barroso said a group was being assembled to assess the impact of the crisis on Europe's economy.
"The volcanic ash cloud has created an unprecedented situation," Mr Barroso said in a statement.
"I have asked (EU Transport Commissioner Siim) Kallas to coordinate the Commission's response and fully assess the impact... on the economy, and the air travel industry in particular."
Long way home
Since Thursday, countries across northern and central Europe have either closed airspace or shut key airports as the ash - a mixture of glass, sand and rock particles - can seriously damage aircraft engines.
Britain has extended a ban on most flights in its airspace until at least 0100 local time on Monday (0000 GMT), air authorities have said.
British Airways cancelled all scheduled short- and long-haul flights for Monday, along with BMI and some smaller airlines.
France said it would close the last airports remaining open at 1200 GMT on Sunday as the ash cloud moved south.
Ireland is closing its airspace until 1200 GMT on Monday.
But Ukraine opened Kiev's Borispol airport, which has been closed since Saturday, and Ukraine International Airlines said it would resume international flights.
Ukrainian President Viktor Yanukovich left for Krakow to attend the funeral of Polish President Lech Kaczynski, a presidential spokesman said.
Many world leaders, including US President Barack Obama, are unable to attend the funeral of Mr Kaczynski, who was killed in a plane crash last week, because of the travel restrictions.
Commuters across northern Europe have sought other means of transport, packing out trains, buses and ferries.
British health officials said any effects of the ash on people with existing respiratory conditions were "likely to be short term".
Southern Iceland's Eyjafjallajoekull volcano began erupting for the second time in a month on Wednesday, sending a plume of ash 8.5km (5.3 miles) high into the air.
Icelandic geologist Magnus Tumi Gudmundsson told the Associated Press news agency: "It's the magma mixing with the water that creates the explosivity. Unfortunately, there doesn't seem to be an end in sight."
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