Scottish Labour, the Scottish Conservatives and the Scottish Lib Dems said Scots did not want another ballot despite SNP claims it had a mandate.
Scottish Labour leader Kezia Dugdale dismissed claims she had not been clear on the issue.
And Ruth Davidson ruled out any second ballot until after 2021.
The SNP has said it has a mandate to hold a second referendum, despite the 2014 vote being a "once in a generation" event.
It said it included the right to hold another independence ballot in the event of a "material" change to Scotland's circumstances.
It insisted that the Brexit vote, in which 62% in Scotland voted to remain in the EU, a majority, albeit reduced at Holyrood last year, and a vote in March by the Scottish Parliament to stage another referendum, constituted a mandate.
Speaking on the BBC's Sunday Politics Scotland Ms Dugdale, who has previously said Labour MPs and MSPs could campaign for independence if a second referendum was staged, insisted she would block any second ballot.
She said: "Everywhere I go I meet people who are distressed, upset and worried about the instability a second referendum would cause and indeed the damage independence would cause.
"We have been very clear we are opposed to independence and a second independence referendum.
"The reality is now, our job is to block a referendum here in Scotland. Because the people of Scotland don't want it, nor do they want independence."
And Scottish Tory leader Ruth Davidson who has seen her party stage an electoral revival on the back of her vocal opposition to another independence vote, said she did not believe the SNP had a mandate for a second referendum.
She insisted on the programme that there was no "public consent" to hold another vote and if there had to be one it could only be after 2021 at the earliest.
She said: "the SNP are going to argue for another referendum till the cows come home. I don't accept they have a mandate. There has got to be both political and public consent for this."
And she said there could not be any second vote held until after a Brexit deal had been negotiated and its effects "played out on the ground" including the new powers devolved from the EU back to Scotland.
She added: "I am absolutely not going to advocate for this to be held in the next while or within the next generation."
And Scottish Lib Dem leader Willie Rennie joined in the chorus against staging a second referendum while campaigning in East Dunbartonshire.
He said: "The Scottish economy is teetering on the edge of a recession, the performance of Scottish education is dropping down the international rankings and mental health services are failing to deliver the care we expect.
"People should use their vote to change the direction of the country away from another divisive independence referendum."
But SNP deputy leader Angus Robertson, said there was only one way for Scotland's "voice to be heard" adding: "Scotland needs MPs who will speak up for what is right, not simply do whatever Theresa May tells them.
"Now more than ever, it is vital to have strong SNP voices standing up for Scotland at Westminster."
Leaders of Scotland's main pro-UK parties insist they would block any attempt to stage a second independence vote.