But according to a poll by Lord Ashcroft, the Nationalists are now set to win the Westminster version of the Edinburgh North and Leith constituency by 14 percentage points at the general election on May 7.
The boundaries are slightly different, but Leith has been represented at Westminster by Labour since time immemorial.
Mark Lazarowicz has been the MP since 2001 when he took over from Malcolm Chisholm, now the MSP.
But he acknowledges he has a fight on his hands. “We know the SNP are making a big push throughout Scotland and it would be surprising if it didn’t affect Edinburgh as well,” says Mr Lazarowicz.
“But our vote has held up better in that poll than in some other places.
“I have said for a long time that, given the rise in SNP support, every seat in Edinburgh is under threat in the same way as elsewhere and we are campaigning very hard. I’m in no doubt this is going to be very close between us and the SNP.
“We are putting our case forward, showing what Labour is doing and highlighting my record as the local MP.”
He says voters do want to talk about Scotland’s place in the UK, but are also worried about issues such as the increase in housebreaking and pressure on GP surgeries. “People recognise these are devolved issues, but people are concerned the Scottish Government should be trying to concentrate on these bread and butter issues rather than what new powers Holyrood should have.”
SNP candidate Deidre Brock, councillor for Leith Walk since 2007 and the Capital’s Depute Lord Provost, says the Ashcroft poll showing the SNP leading Labour by 14 points – 43 per cent to 29 – was encouraging and chimes with the party’s own canvassing returns.
But she says: “We’re not taking anything for granted and we’re speaking to as many people as we can to get the SNP message across.”
Councillor Brock – originally from Australia and once an actress in Home and Away – says voters want change.
“The main thing people say is they think it’s time for a change in Westminster politics. They are tired of the same old tango between Labour and Conservative.
“There’s a real feeling of excitement in the area, which I find very encouraging.
“People genuinely feel this time there might be a chance for real change – that’s the message I’m getting at least.
“After the referendum there is a recognition that things cannot stay the same.”
And she says that the level of interest during the referendum has lasted into this election.
“People are so engaged it’s incredible,” she says. “Back in the day, you would go to a door and some people would be almost apathetic and they didn’t want to talk about politics.
“Now practically everyone wants to talk about it – and that’s true whichever side they were on in the referendum. That’s a sea-change for me.”
Tory candidate Iain McGill fought the seat last time and says he notices a big difference between then and now.
“The Labour vote is down, the Lib Dem vote is disappearing, our vote is up and the SNP vote is up.”
And he claims the Conservatives are upbeat. “We have a positive record in government which we are not ashamed of. The Lib Dems are trying to get away from their record in government and Labour are trying to get away from their record in the referendum. We are the only unionist party whose vote is going up.
“This constituency voted 60-40 No in the referendum, well above the national average. It has not been a hotbed of nationalism.”
He says people have been asking if they should vote tactically. But he says: “We tell them that whether they vote Labour or SNP what they will get is Labour controlled by the SNP.”
In the last three elections, the Lib Dems have been second in this seat. But candidate Martin Veart acknowledges everything is different this time.
He says the picture is mixed across the constituency. “I did a hustings the other night and I don’t think I got many votes, but there are other areas which are much more sympathetic.
“We are still out there and we’re still fighting. I’m doing my best to make sure the Liberal Democrats are fully represented.”
Green candidate Sarah Beattie-Smith says she is getting a “really positive” reception.
“I’ve been pleasantly surprised by the number of people who say they have always agreed with Green Party policies and this time they’re going to vote for what they believe in.”
Ms Beattie-Smith, who works for Citizens Advice Scotland, says people have a wide range of concerns. “They talk about the economic situation and austerity and the impact it’s having on people, but also about Scotland’s place in the world, especially after what’s been happening in the Mediterranean.
“But after the referendum, when people found their voice, they’re frustrated it’s now back to the same old Westminster politics.”
Journalist Bruce Whitehead is standing for Left Unity, the anti-austerity coalition launched by film director Ken Loach.
He says: “The main four parties have had decades to get it right and they’ve utterly failed. With the SNP in charge, we now have half-a-million in poverty, low wages, cuts in essential services, while they build more roads and bridges we don’t need. Meanwhile, the wealthy get wealthier, avoid their taxes, and governments fight wars over oil and territory to enrich corporate backers.
“People who vote SNP to punish Labour are just replacing MPs from a London-based capitalist party with MPs from an Edinburgh-based capitalist party, which is not going to be any better.”
Edinburgh North and Leith stretches from Leith Links through the heart of Leith to Trinity, Granton and Pilton, takes in the New Town, Broughton, Warriston, Inverleith, Stockbridge and Dean Village.
A lot of new house-building in the seat has seen a big increase in the electorate since the last election.