The latest PA general election projection also had the Conservatives well ahead with 381 projected seats and Labour on 181.
But First Minister Nicola Sturgeon hit back, branding claims the Tories were a threat to the SNP "ludicrous" and that Miss Davidson risked being left with "egg on her face". The campaign is barely a week old and if the SNP message is already looking shop-soiled that's because it is.
Scottish Labour, a party that used to weigh its vote in town halls across Scotland, particularly in the West, did predictably badly, now finding itself where the Scottish Tories once languished, unloved and increasingly irrelevant. The difficulty is that if...
Still, that disappointment did not stop the SNP winning half the vote in the United Kingdom election in 2015, so perhaps it will not do so again. "We will not be a government that presides over food bank Britain or asks teachers to go on the school gates and collect funds to pay the teachers' wages", he said.
Labour was beaten into third place by the Conservatives in Scotland, where the SNP was comfortably the largest party despite failing to take control of target councils.
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Nicola Sturgeon will be in Perth on Monday to tell voters the SNP will "deliver for Scotland".
Writing in The Mail On Sunday, after her party won a record 276 councillors north of the border, more than double the number they secured five years ago, Ms Davidson suggested Scots are "fighting back against the SNP's independence obsession".
Liberal Democrats improved by one point to 9% and Ukip were unchanged on 7% - however the figure was released before the humiliating losses during local elections. Download it today and continue to enjoy STV News wherever you are.
"We won the local government election in Gordon this week, beating the SNP into second place".
Party insiders said the Scottish Conservatives leader was "parking her tanks on Alex Salmond's lawn" after ousting an SNP councillor in Inverurie.
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But matters could be complicated by the lack of overall majorities, and complex talks are now getting under way across Scotland to try to stitch together either formal coalitions or informal deals to give councils workable administrations. Scottish Conservatives even have their first ever councillor in the Western Isles.
Labour's former council leader Frank McAveety, who retained his seat in Shettleston, said: "The SNP predicted they could return 56 councillors, they've only returned 39 councillors".
The analysis also suggests that the party is ahead in Edinburgh South, which is now home to Labour's only MP in Scotland, Ian Murray.
Sturgeon rejected the suggestion that the SNP's share of votes indicated it would struggle to achieve a majority in a future independence referendum, insisting the two contests were not comparable. And the SNP have not had any difficulty increasing their vote in local by-elections during the last two years.
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