The airports operator BAA today formally dropped plans for new runways at Heathrow and Stansted airports, bringing to a close one of the most controversial parts of the last Labour government's transport policy.

Local campaigners, who spent eight years arguing that the expansion would not only result in the demolition of hundreds of homes but wreck the peace of thousands more with extra aircraft noise and pollution, said they were overjoyed at the news.

"My first reaction is – hooray! I can hardly believe it. I can't believe that we've finally won. My first thought is that we need to plan a party," said Geraldine Nicholson from the No Third Runway Action Group, based in villages to the north of Heathrow's perimeter.

Since the expansion proposals were outlined in a government paper in 2002 BAA, which operates both airports, has pressed ahead with plans for both sites, buying up a number of properties in preparation.

However, both the Conservatives and Liberal Democrats came out against the schemes, and with Labour looking set to leave office, BAA said late last year it was shelving its programme at Heathrow until after the election.

The new government's formal coalition agreement last week promised to cancel Heathrow's third runway and rule out a new runway at Stansted, as well as at London's third main airport, Gatwick.

Following this announcement, BAA today said it would stop all work on a planning application for the third Heathrow runway.

In a separate statement, it said it was also withdrawing its plans for a second Stansted runway. Schemes to buy houses in both affected areas would also end, it added, although some purchases already underway would continue.

The company's chief executive, Colin Matthews, said: "We recognise the importance of government policy in a matter as significant and controversial as runway capacity.

"The policy intentions of the new government are clear and it is no longer appropriate for us to purchase properties. Equally important is our intention to stand behind our commitment made to provide time for those local residents who wish to sell their properties to us, to do so."

Speaking of Heathrow, he added: "We continue to believe that new capacity would strengthen the UK's trading links with the global markets on which our economy and our competitiveness depend."

The Heathrow expansion would have resulted in the demolition of more than 700 homes in the village of Sipson, along with its school, church, pubs and restaurants to make way for the runway.

Nicholson said the village, and surrounding areas, had been badly affected by its potential fate: "The uncertainty has been the hardest part. For the whole area it's been like living under a black cloud. Now that black cloud has finally been lifted. It's fantastic."

Carol Barbone, from the Stop Stansted Expansion group, said: "We are absolutely delighted. We are rejoicing, thrilled. It's been eight long years. We are nonetheless mindful that things can change and we will be looking for a longer term commitment on this.

"It's not something we expected. BAA have been trying religiously to build this runway. If they are able to roll over so easily now, why couldn't they have done it before?"