Officials in several Ukrainian cities have refused to accept the outcome of Sunday's presidential election.
Yushchenko supporters held rallies in the city of Lviv as well as Kiev
Tens of thousands of protesters have rallied to contest the official victory for Prime Minister Viktor Yanukovych, amid Western concern over the vote.
Opposition challenger Viktor Yushchenko has told supporters to stage a civil disobedience campaign.
But central security authorities are warning that they are ready to put down any lawlessness "quickly and firmly".
"We appeal to the organisers of mass protests to assume responsibility for their possible consequences," the prosecutor general and the interior ministry said in a statement.
The central electoral commission said that, with more than 99% of the vote counted, Mr Yanukovych had 49.4% of ballots while Mr Yushchenko had 46.7%.
But the opposition says it has recorded many thousands of irregularities - including very high turnouts in government strongholds.
By late evening on Monday, thousands of opposition supporters had left Kiev's Independence Square after demonstrating for more than 12 hours. But several hundred people planned to spend the night in tents in the area.
The opposition told people come back on Tuesday morning for a protest outside parliament, when MPs are due to discuss the contested election result.
The US state department said it was "deeply concerned" about the election and threatened to review its relations with Ukraine if the government failed to investigate the allegations of election fraud.
"Should, in the final analysis, this election prove to be fundamentally flawed and tarnished, we would certainly need to review our relations with the Ukraine and consider further steps against individuals who had engaged in fraud," spokesman Adam Ereli said.
Mr Yushchenko, seen as the pro-Western candidate, earlier told his supporters in the capital not to leave their rally "until victory".
"We are launching an organised movement of civil resistance," he said, denouncing what he called the "total falsification" of the vote, which followed days of acrimonious wrangling over the results of the first round.
Imprisoned twice in his youth
Former governor of industrial Donetsk region
Raised pensions and public sector pay before election
Would make Russian second official language and allow dual citizenship
An economist and former central banker
Has an American wife
Promises to fight corruption, create five million jobs and pursue free market reforms
Would seek deeper relations with Europe and the West
Kiev city council refused to recognise the results, and urged parliament to follow suit.
Thousands of people also turned onto the streets in the western city of Lviv, where the city council said it would only take orders from Mr Yushchenko.
Three other cities in opposition strongholds in western Ukraine have said they considered the opposition candidate the legal president.
The city councils' move is likely to be seen as a symbolic moral victory for the opposition - although the councils have much less power than the central authorities, observers say.
Mr Yanukovych has called for national unity and criticised the call for public protests.
"This small group of radicals has taken upon itself the goal of
splitting Ukraine," he said in comments reported by AP news agency.
Observers for the Organisation for Security and Cooperation in Europe (OSCE) said Sunday's run-off vote fell far short of European democratic norms.
The organisation, which also reported serious irregularities in the first round, said violations included a continuing "media bias" in favour of Mr Yanukovych and intimidation of observers and voters.
The US' official observer, Senator Richard Lugar, alleged "concerted and forceful" fraud and the EU called on Ukraine to review the election.
However, Moscow, which backed Prime Minister Yanukovych, recognised the result.
Exit polls earlier suggested that Mr Yushchenko had been on course for victory with a lead of at least six percentage points.
His supporters say they do not believe the official turnout figure of 96% in eastern Ukraine.
During the campaign, Mr Yushchenko, prime minister between 1999 and 2001, claimed to have been the victim of intimidation and dirty tricks, including an alleged poisoning attempt.
His critics portray him as an American puppet who will do anything to gain power, including inciting civil unrest.