She added: "I certainly have no qualms or any fear about governing as a minority administration, and talking to colleagues from other parties on an issue-by-issue basis to get support from them".
Labour and the Lib Dems said they would target seats in a bid to oust SNP MPs.
Ms Sturgeon said: "Yes, the Tories made gains and had a good performance by their standards - but that support came from Labour not the SNP, so Labour and the Tories are fighting it out for second place while the SNP continues to be comfortably in first place".
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".
Voters in local and mayoral elections took to the polls in many parts of the United Kingdom on Thursday.
The main one is Robertson's healthy 9000 majority (though the SNP of all parties should know that no majority is safe when a tide has turned having wiped out bigger Labour leads in 2015), bolstered by his increased profile since the SNP became the third party in the Commons and cemented by the fact he is a brilliant campaigner who has masterminded most of the SNP's success at the ballot box in recent years.
Ms Davidson said the results showed "only the Scottish Conservatives have the strength to fight back against the SNP".
A total of 1,227 councillors are being elected across the 32 councils.
"There are 10 Tories and two independent members, those two independents may decide they'd prefer to work with Labour".
"Across Scotland, there has been a clear backlash against the SNP's plans for a divisive second independence referendum and anger over the SNP's woeful record running our schools, hospitals and public services".
The poor set of results included Labour losing councils that it had previously controlled outright, but Kezia Dugdale's party came first in three local authorities and has the potential to be kingmaker elsewhere.
Ms Sturgeon said: "I'm over the moon that we are the largest party in Glasgow and are about to form an administration in the city".
He said: "I'm shocked... when you stand as a candidate for the Conservatives in the east end of Glasgow you don't expect something like this to happen, but I think people recognised I'm a local voice and will stand up for local issues".
She 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.
What Nationalist strategists should - and probably will - be worrying about, however, is whether Scottish politics is re-balancing away from their party.
No longer toxic, it would seem, the party is gaining seats in every part of the country.
May will also be watching Scotland to see how far her party's revival extends in an area where it had suffered years in the political wilderness.
The latest results place the SNP on 24, Labour on 19, and the Greens and Tories both on six.
After further declarations, the SNP needs to win 19 of the 30 seats left to be able to take overall control of the council.
Green co-convener Patrick Harvie said Labour voters were "almost apologising" for their party choice on the doorsteps, and he predicted no one party would have overall control of the council.
"East Dunbartonshire, Argyll, Caithness and North East Fife were superb".