The operation to unload more than 2,000 containers from the MSC Napoli a mile off the Devon coast is under way.
The salvage operation could take about five months
Some 3,500 tonnes of oil is being pumped from the ship, the Maritime and Coastguard Agency said.
The Napoli, which was hit by storms, was deliberately grounded 10 days ago amid fears it would sink in deep water.
Devon County Council has said it will hold an inquiry into why the ship was grounded off a World Heritage Coast if the government does not investigate.
A Devon spokesman said there was "overwhelming support" among members of the authority's executive to hold a local inquiry if necessary.
The Royal Society for the Protection of Birds has estimated that more than 1,600 birds have been oiled as a result of the Napoli incident.
Nearly 100 miles of the Devon and Dorset coast have been surveyed by RSPB staff and volunteers in search of birds oiled as a result of spilled fuel from the vessel.
Mark Clark of the Maritime and Coastguard Agency said half a tonne of oil which escaped on Sunday was being dealt with.
The operation to remove the containers failed to go ahead on Sunday after a mooring cable snapped on one of the vessels carrying a crane.
A team of about 20 salvage operators will be involved in the removal of the containers, some of them divers who will go into the ship which has been flooded with water and oil.
The divers will be working in a dangerous environment when they move around the precariously positioned containers to find the best ways of getting them off.
Mr Clark said they would have to take the "greatest of care" with such a dangerous work area.
"The vessel is at an acute angle with crushed containers and spilling goods. People have to get on those containers and secure heavy chains so it's a very dangerous job," he said.
"Their lives are more valuable than the cargo."
The salvage operation could take about five months as the team are working at a protected World Heritage Site.
The largest barge crane will remove the 2,291 containers - laden with everything from BMW motorbikes to nappies - in priority order.
The smaller crane will then transfer them to a barge which will take them to Portland port in batches of up to 90 at a time.
The MCA has warned opportunists they face arrest if they take goods that wash up ashore while the salvage operation is under way.
At a public meeting for 300 Branscombe villagers on Saturday some locals complained about the way the situation has been handled. The police have defended their actions.
The 62,000-tonne MSC Napoli, which was holed in storms on 18 January, was deliberately run aground because of structural damage.
MSC NAPOLI SALVAGE OPERATION
The Napoli's containers will be unloaded from the stern first by crane barge Big Foot
A second crane on Big Foot will then transfer containers to shuttle barge Boa Barge 21
Shuttle barge will then transfer containers to Portland Harbour, where they will be offloaded
Lightering vessel Forth Fisher is positioned on the other side of the ship pumping off its remaining fuel oil
A decision will be made over whether to re-float the ship after all the oil and containers have been removed
Key fuel tanks and total contents identified by the Maritime and Coastguard Agency on 23 January. Oil is being pumped off at a rate of approx. 30 tonnes per hour