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Brian Currie Political Editor
Labour not only held on, but regained overall control of the city which the SNP said would be its "stepping stone" to independence.
In a day where both parties made significant gains at the expense of the LibDems, Labour retained power in North Lanarkshire, now leads the administration in Renfrewshire and West Dunbartonshire, has most councillors in Edinburgh and overall is the biggest party in 14 councils.
The SNP took overall control in Dundee and Angus, and has more councillors than any other party – a total of 424 after adding 57 to the number going into the election. It is also the largest party in 10 local authorities. Mr Salmond said it was a "substantial achievement" and a "major step forward for the SNP and for Scotland".
However, Labour is close behind with 394 councillors after increasing its number by 58.
After the victory in Glasgow, Labour's Scottish leader Johann Lamont said: "If last year was a tsunami for the SNP, perhaps now the tide is going out on Alex Salmond."
The biggest losers were the LibDems who lost 80 councillors, leaving them with just 71.
They performed so badly that in one Edinburgh ward they were outpolled by a candidate dressed as a penguin – "Professor Pongoo" Mike Ferrigan, an independent climate activist. LibDem council leader Jenny Dawe lost her seat, with many voters blaming her administration for the trams fiasco. Many spoiled ballot papers had "trams" scribbled on them. Scottish LibDem leader Willie Rennie admitted it had been a "very distressing day".
The Conservatives dropped by 16 to 115 councillors but 14 Green candidates were successful – five of them in Glasgow where they increased their share of the vote, pulling ahead of the Tories and LibDems.
A spokesman claimed they were "now clearly the third party of Glasgow and the only credible alternative to Labour and SNP".
The SNP celebrated forming its first majority administrations under the single transferable voting system in Dundee and Angus councils. In Dundee, every one of its 16 candidates was elected.
Mr Salmond claimed his party was the overall victor, saying: "With over 420 councillors, an increase of around 60 on 2007 and double the lead over Labour the SNP has won Scotland's election. Five years after backing the SNP for the first time Scotland continues to move forward with the only national party. That is a substantial achievement.
"The SNP has won seats from Labour, from the LibDems and from the Tories in all parts of Scotland, urban and rural. We set ourselves the target of securing more councillors and we have met that target with around 60 extra councillors."
He also claimed the council elections were a "tale of two governments", contrasting the fortunes of the Coalition at Westminster with his Holyrood administration.
Mr Salmond said: "The LibDems and Tories have had a disastrous day, feeling the full force of the Scottish people who have rejected their damaging austerity agenda in favour of the SNP locally and nationally."
Ms Lamont was equally upbeat claiming it had been a "very good day". She added: "Last year the Scottish people sent us a clear message that we had to up our game, that we had to put their interests before the party's interests.
"We have taken a major step forward in rebuilding faith. I was elected leader of this party on a mandate of change and the results show that we are now making progress on the path of renewal.
"Results like Renfrewshire, where we have a majority after the SNP controlled the council, were remarkable as were results in North Lanarkshire, East Lothian and Dumfries.
"We are back on the pitch and fighting for people and their priorities."
Labour's UK leader Ed Miliband said the Glasgow result showed his party was "beating back the SNP challenge".
In 2007 the LibDems were the third party of local government with 166 councillors, and the party was involved in 13 coalitions across Scotland.
Mr Rennie, who took over as leader after they suffered heavy losses in last year's Holyrood election, said they had "lost many strong community activists, who have stuck up for their area for many long years".
While both the LibDems and the Tories endured a bruising night south of the Border, Mr Rennie said the results "should dispel any myth that the Liberal Democrats are only in the Coalition for ourselves".
He insisted the coalition with the Conservatives at Westminster "has always been about doing the right thing for the fortunes of the country".
He added there were "many strong LibDem councillors who will join the rebuilding process for the party".
Tory leader Ruth Davidson also remained positive despite her party returning fewer councillors than in 2007.
She claimed "a number of gains in councils across Scotland" and said: "We are now heading towards becoming the third party of local government in Scotland for the first time since 1992.
"We are the largest party in the Borders and South Ayrshire, and we will play a pivotal role in forming a number of administrations across Scotland."
Talks between the parties in areas where there is no overall control will take place over the next few days as they seek to form alliances.
To see the results from every ward in Scotland, go here
Mr Miliband claimed Labour had shown voters it was "back throughout the country on your side" after making big gains south of the Border, taking control of 29 councils.
It was the party's best night at the polls since Tony Blair's leadership, making inroads into the south of England.
It seized key councils such as Southampton, Exeter, Plymouth, Reading and Harlow and even snatched seats in David Cameron's Witney constituency in Oxfordshire.
Disgruntled Tory MPs called on Mr Cameron to assert Tory pre-eminence over the LibDems and ditch "barmy"policies on gay marriage and reform of the House of Lords.
