No-one was more surprised than Labour at their success in Glasgow in the local elections.

Their spokespeople were still pinching themselves as they headed to the television studios on Friday night. They'd already prepared their excuses: disgraced former leader, party divided, turmoil of transition. Even some Labour people in Glasgow think that they've has been in power too long and could do with a shake-up, which is what proportional representation is supposed to do by preventing councils like Glasgow remaining one-party states.

What no-one expected was that the voters would opt to keep the one-party state. And there's no getting away from it, Glasgow was a remarkable result for an election conducted on the Single Transferrable Vote. Labour ended the night with a commanding overall majority on the council, 44 seats to the SNP's 27, showing that there is life yet in the people's party.

The SNP had fallen for its own propaganda that Labour were finished here. Labour's battle-scarred council leader, Gordon Matheson, declared that the "SNP juggernaut has landed in a Glasgow ditch" – arguably repeating the mistake made by his SNP counterpart, Allison Hunter, whose presumptuous remark that Glasgow was "a stepping stone to independence" is thought to have damaged the Nationalist case.

SNP members were incandescent at newspaper headlines about their "bloody nose". We won fair and square across Scotland! they cried. Look at our record tally of 424 seats overall; our increased voting share; look at Dundee, look at Angus! One slightly hectic Nationalist commentator tweeted that the SNP were now "the third party in the United Kingdom" because they'd won more seats in Scotland than the Liberal Democrats in the UK. The Twittersphere fizzed with twits condemning the bias of the BBC and the unionist press. Professor John Curtice, the noted commentator, briefly became a target of the CyberNats for suggesting that Labour might have done better on first preference votes than the SNP.

In fact, it was a game of two halves for the Nationalists – a tale of two cities, even. In striving, aspirational Dundee, which considers itself the heart of the new Scotland, they swept the board; whereas in cautious, welfare-dependent Glasgow they were rejected in favour of the Labour security blanket. With the Conservatives in power in Westminster, Scottish voters in traditional Labour areas like Fife and Renfrewshire were looking for more than lectures on independence and responded well to Labour's mantra of jobs, jobs, jobs. The SNP needs to remember that it does best when it out-Labour's Labour.

The Murdoch affair probably played a part too. The SNP had hopes of becoming the largest party in Edinburgh, where even penguins were outdoing the Liberal Democrats. But a lot of middle-class voters who had flirted with the SNP in 2011's Holyrood elections have been appalled at Alex Salmond taking tea and Tunnocks Caramel Wafers with the hated Rupert Murdoch. It's likely many of them sought refuge in Labour, which became the capital's largest party.

Again, the SNP needs to remember that it won last year's Holyrood landslide essentially because of its performance in government. The Nationalists are vulnerable whenever they appear to be behaving like the other big parties. The support of the Sun newspaper has been bought at a high cost.

But the local elections were in no way a disaster, or even a setback for the SNP in Scotland – it just wasn't another landslide. They make solid progress even in Glasgow, where they gained five seats on their 2007 showing. The circumstances weren't right for a colossal switch to the Nats, and perhaps there is a crumb of comfort here for the SNP. For a party that has been in government for fully five years, this is a considerable achievement. On this showing, Scotland is still pretty content with Alex Salmond.

You could almost argue – and some will – that this was a good basis for the referendum campaign, which starts later this month. The voters who backed the SNP last week were doing so fully aware that they were voting for a party committed to and working towards independence.

The SNP believe that Scotland is becoming polarised on the constitution and that their voters are becoming "harder" for independence. I'm not so sure of this myself, but I can see the argument. Across Scotland people are getting used to voting SNP – just as they were used to voting Labour or Tory in the past. Perhaps Alex Salmond will be able to persuade them that voting for independence is natural, too, come October 2014.

But the danger with referendums is that people rarely vote for the question on the ballot paper. Their general attitude to the government colours their preferences, particularly if the government of the day has upset them.

And the SNP has started to upset people. Something about Salmond being a bit too pleased with himself, a bit too autocratic, a bit too hugger-mugger with a particular press baron. Labour has been banging away at the SNP for being the Tories' little helpers in Scotland to little avail, but Johann Lamont seems to have struck home with her recent attacks on the First Minister's gullibility when dealing with the world's Trumps or Murdochs.

However, it may all fade away after Salmond's appearance before the Leveson Inquiry, and this is not the kind of issue that makes much of an impact in the streets of Glasgow. But if Labour continue to harp on about the SNP's monomania with independence while Scots are losing their jobs and homes in a double-dip recession, they may be able to do lasting damage.

The SNP is no longer in its honeymoon phase when it was pleasing Scottish voters by saving accident and emergency units and abolishing prescription charges. The Nationalists need to develop an agenda for this government that does not begin and end with independence, and that may not be so easy.

The SNP's very party discipline could become a disadvantage here. People are becoming bored by rent-a-quote SNP spokespeople parroting the leadership line, just as they became bored with the Labour "pager clones" back in the late 1990s. Scotland is a robust political culture; it doesn't look good for the Nationalists to be acting like lobotomised servants of a North Korean leader. Even New Labour had its Tony Benns and Dennis Skinners.

And a bit of dissent in the Nationalist ranks might be no bad thing right now, because people are beginning to worry about Salmond having too much control over the party, parliament and the country. The First Minister is still the SNP's best asset, a gifted politician with a unique ability to connect with ordinary people while putting the intellectual case for independence with style and substance. But maybe even Alex Salmond needs to be saved from himself.