They tabled a confidence motion in parliament which the government lost by a single vote.
But the opposition is threatening to move a no-confidence motion in the government to force an early poll.
The opposition leader, Mr Sharif, holds the cards: any no-confidence motion in parliament would need his backing to succeed.
The coalition government of Jean-Luc Dehaene survived a no-confidence motion in parliament.
Aubrey Jewett, a political science professor at the University of Central Florida who specializes in Florida politics and state and local government, said Triplett's vote Monday to reject the police chief's resignation was surprising since the mayor voted for the no-confidence motion in the first place.
If all of Mr Cameron's opponents were to unite against him over, say, his finance measures or in a confidence motion, they could force another general election.
The SNP want him to resign as chair of his select committee - but have discovered to their chagrin that the rules require that any no-confidence motion put in the Committee would have to be signed by at least two members of the largest party on the Committee - ie Labour members.
In March, Sanford's city commission passed a 3-2 motion expressing "no confidence" in the police chief.
Last week, the ATL teachers' union also passed a motion of no confidence in Mr Gove.
The motion of no confidence in Lesley Griffiths was defeated by 29 votes to 28.
Conwy council has passed a motion of no confidence in the Betsi Cadwaladr University Health Board.
BBC: North Wales NHS: Betsi Cadwaladr no confidence vote passed
Sanford's city council passed a motion of no confidence in the city's chief of police, who is now suspended.
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The National Union of Teachers' conference has unanimously passed a motion of no confidence in the Education Secretary, Michael Gove.
Teachers have passed a motion of no confidence in Education Secretary Michael Gove and the chief inspector of England's schools, Sir Michael Wilshaw.
The NUT conference began a motion of no confidence in Michael Gove - which was adjourned before it could be put to a vote.
Meanwhile, a motion of no confidence in Donegal county secretary, Noreen Doherty tabled by Setanta hurling club has been referred upwards to Croke Park.
The presiding officer of the 60-member National Assembly cannot vote in a no-confidence motion, leaving Mr Michael with 28 votes to his opponents' 31.
On 9 October last year, Mr Weir did not vote against a DUP motion of no confidence in the first minister, party leader David Trimble, it added.
His conciliatory words come after delegates passed a motion of no confidence in Education Secretary Michael Gove and the Chief Inspector for Schools in England, Sir Michael Wilshaw.
But a spokesman for Mr Cameron said later that the motion of no confidence in the Speaker should be debated, if it got enough signatures in order for his future to be resolved one way or another.
Under the legislation, a general election could only be called early if two thirds of MPs backed a motion of no confidence in the government or if a new administration could not be formed within 14 days of a majority of MPs backing a no confidence resolution.
Opposition parties united behind a motion of no-confidence in Mrs Griffiths in a row about a controversial report into reorganising hospitals.
The first agreed motion asked if they had lost confidence in Mr Richards and if they felt their relationship with him had irretrievably broken down.
The motion to hold a no confidence vote in Mr Whalley was defeated, but then the whole process was repeated just a few months later in early 2009.
The Treasury had been hoping to inspire that confidence when, earlier this week, it set in motion the largest government intervention in the American banking system since the Great Depression.
But having failed to mount an effective no-confidence motion against Mr Obuchi's government in the all-important lower house of the Diet (parliament) for its handling the country's banking crisis, Mr Kan and his opposition colleagues have now moved their guerrilla campaign to the upper house where they have a majority.
In moving his party's motion on car-jacking, Mr Maskey said the public had lost confidence in how the police and judiciary were dealing with the crime.
He said a vote in parliament on Wednesday, on a no-confidence motion tabled by the Socialists, had "re-legitimised" the government.
The chronic debt problems, coupled with the political gridlock in the United States, have had a significant impact on confidence, and confidence is what we need to get and keep the global economy in motion.
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