Labour will not "frustrate" the process for invoking Article 50, Jeremy Corbyn said after the Supreme Court ruled that Parliament must give permission to start the formal mechanism for Brexit.
But the Labour leader said his party would seek to amend the legislation in order to prevent Theresa May allowing the UK to become a "bargain-basement tax haven".
The SNP has vowed to table 50 amendments to any legislation, while the Liberal Democrats have said they will oppose the triggering of Article 50 unless there is a second referendum on the final deal.
Mr Corbyn said his party wanted the final deal to be put to a "meaningful vote" in Parliament instead of a plan which would see the proposed deal presented with no option for renegotiation.
He said: "The Government has today been forced by the Supreme Court to accept the sovereignty of Parliament.
"Labour respects the result of the referendum and the will of the British people and will not frustrate the process for invoking Article 50.
"However, Labour will seek to amend the Article 50 Bill to prevent the Conservatives using Brexit to turn Britain into a bargain-basement tax haven off the coast of Europe.
"Labour is demanding a plan from the Government to ensure it is accountable to Parliament throughout the negotiations and a meaningful vote to ensure the final deal is given Parliamentary approval."
The SNP, the third largest party in the Commons, will put forward dozens of "serious and substantive" amendments to the legislation.
Former Scottish first minister Alex Salmond, the SNP's international affairs spokesman, said: "We welcome the Supreme Court's decision and hope that their ruling brings this Tory Government back to the reality that they cannot simply bypass elected parliamentarians to fulfil their role in carrying out due and proper scrutiny of one of the biggest decisions facing the UK."
He said the SNP's amendments would ensure that the devolved administrations are treated as "equal partners".
"If Theresa May is intent on being true to her word that Scotland and the other devolved administrations are equal partners in this process, then now is the time to show it.
"Now is the time to sit with the Joint Ministerial Committee and not just casually acknowledge, but constructively engage. Consultation must mean consultation.
"Our amendments will address the very serious concerns facing the UK and the very real issues that the UK Government has, thus far, avoided."
Lib Dem leader Tim Farron said: "This Tory Brexit Government are keen to laud the democratic process when it suits them, but will not give the people a voice over the final deal. They seem happy to start with democracy and end in a stitch-up.
"The Liberal Democrats are clear - we demand a vote of the people on the final deal and without that we will not vote for Article 50."
Green Party co-leader Caroline Lucas confirmed she would vote against triggering Article 50 on the timetable set out by the Prime Minister.
Mrs May has vowed to trigger Article 50, starting the formal two-year countdown to Brexit, by the end of March.
Ms Lucas said: "The spotlight now falls on MPs - and in particular the Labour Party - to properly scrutinise the Government's plans and act accordingly.
"That must mean that Labour rethink the support they've given to triggering Article 50 prematurely, and instead join those of us who refuse to be pushed into Theresa May's artificial Brexit timetable."
Ukip's sole MP, Douglas Carswell, said the ruling cleared the way for Mrs May to trigger Article 50.
"All the lawyers in London are not going to be able to stop us leaving the European Union," he told Sky News.
"We need two votes - one in the House of Commons, one in the House of Lords. If either of those chambers attempt to veto the will of the people, we can replace the Commons with a general election and we can certainly replace the Lords by creating 800 new peers."
Ukip leader Paul Nuttall said: "This decision is hardly a surprise but in the end it will make no real difference.
"The will of the people will be heard, and woe betide those politicians or parties that attempt to block, delay, or in any other way subvert that will.
"Other than making clear that this is a decision of the whole United Kingdom, rather than its constituent parts, what we can clearly see is that it will embolden those who rail against the decision of the people.
"It may give heart to those in the EU, used as they are to ignoring their own people, to attempt to play hard ball in the negotiations.
"But in the end I am convinced that, though this skirmish has been lost in the courts, the war will be won".
Pro-Brexit Labour MP Kate Hoey, of the Labour Leave campaign, said the party risks losing the upcoming parliamentary by-elections in Copeland and Stoke-on-Trent Central if it seeks to block Article 50.
"If Labour MP's are seen to be frustrating the will of the people by opposing Article 50 then they will lose their seats," warned Ms Hoey.
"It is time for the Labour Party to support the Government by voting for Article 50 and working together to ensure the United Kingdom enjoys the global opportunities Brexit provides."
Labour Leave chairman John Mills - a major party donor - said: "If Labour is to stand a chance of winning in 2020 or 2025 it needs to win over Tory and Ukip voters, the overwhelming majority of whom voted to leave the European Union.
"If we continue to flap about on this issue instead of getting on with making a success of Brexit the voters will not forgive us.
"It is time for the Labour Party to unite around Brexit and present its own vision for Britain outside the European Union."
Prominent Brexiteer and Conservative MP Dominic Raab, a supporter of the Change Britain campaign, said: "The Government was right to appeal, and the Supreme Court has usefully made clear that a short Bill authorising the start of negotiations is all that is needed.
"Let's have an end to the wrecking tactics.
"Every democrat in Parliament should now support this legislation, so we can get on and deliver on the will of the British people, and secure the best possible deal for the whole country."
MP Anna Soubry, who was one of the most vocal Conservative advocates of a Remain vote, said: "Our independent judiciary are to be congratulated on their wise judgment. Parliamentary sovereignty protected."
Labour MP John Mann said: "The Supreme Court has been very clear.
"The Government must now publish the Brexit Bill and I expect it to receive overwhelming support in Parliament.
"A straightforward Brexit Bill will sail through Parliament and it is time for the Government to get on with this."
Martha Spurrier, director of civil liberties group Liberty, said: "Our democracy hinges on two principles: no one is above the law and Parliament is supreme. Today's ruling upholds those principles.
"Thanks to our independent judges, Mrs May's Government will be exposed to the antiseptic of parliamentary scrutiny.
"This is not a political decision - it is our democracy in action."
Foreign Secretary Boris Johnson said in a tweet: "Supreme Court has spoken. Now Parliament must deliver will of the people - we will trigger A50 by end of March. Forward we go!"
Lord Newby, leader of the Liberal Democrats in the House of Lords, said it was not the role of peers to vote down the Bill.
He told Sky News: "I don't think the House of Lords is going to seek to stop this Bill from being enacted within the due time and the Government will then start its negotiations.
"But one of the few agreed roles of the House of Lords is that if it thinks the Government and the House of Commons has got something wrong it can ask them to think again."
Leave.EU chairman Arron Banks said: "Today's judgment gives our out-of-touch establishment the ability to soften or delay the clean Brexit a majority of the British people voted for.
"The people have been let down. Parliament gave us a referendum and the people had their say, yet the power has now been handed back to Westminster by our unelected establishment judges. This decision shows how broken the system is - true democracy is being thwarted.
"This Tory Government wants to be a champion for ordinary people, now May must prove it by driving Brexit through Parliament post-haste - 17.4 million voters will be watching her every move."
A Welsh Government spokesman said: "We welcome the Supreme Court judgment today which means that the UK Government must seek parliamentary approval in the form of a Bill before it can trigger Article 50 to leave the European Union.
"It is vital that the UK Government develops an approach to negotiations which reflects the interests of Wales and the UK as a whole - an approach to which the Prime Minister has already publicly committed.
"The judgment preserves and recognises the importance of the Sewel convention whereby Parliament will not normally legislate on areas devolved to Wales and devolved governments without their consent.
"The Welsh Government will continue to work closely with the UK Government, and the other devolved administrations, through the Joint Ministerial Committee process to influence the overall UK position.
"Our aims are to preserve single market access for business and to protect Welsh jobs and investment, along with the rights of workers."
Dick Newby, leader of the Liberal Democrats in the House of Lords, said the ruling was not a surprise.
He told the Press Association: "The Government was always on very thin legal ice and on hopeless political ice in trying to deny Parliament the chance to trigger Article 50, so it's a sensible decision by the judges so Parliament can get on with it.
"I think what's going to happen now is we are going to look to see the extent to which the Bill itself can be amended, and in the case of the Liberal Democrats we're going to try and insert into the Bill a provision that there should be a referendum on the final terms of the deal, because the British people certainly didn't vote for the hard Brexit that Theresa May seems to be pursuing."
Lord Newby said that a new referendum would be a first referendum on the actual terms of a Brexit deal.
He said: "People voted for all kinds of reasons last June, but they certainly didn't vote in large numbers for the deal that Theresa May now seems that she is going to negotiate.
"So we think it's only reasonable that the people kicked off the process, the people should have the final decision."
Tory former chancellor Lord Lamont said the split vote showed the law is "not unambiguous" but warned against delaying Article 50.
"We are about to discover which MPs, which peers and which parties are sincere in saying they respect the vote of the people," he said.
"Any delay will lead to an enormous awakening of bitterness.
"The House of Lords in particular, as an unelected body, needs to tread very carefully and avoid triggering a constitutional crisis."
Tory former Cabinet minister Iain Duncan Smith said the ruling was made by a "self-appointed court".
He told BBC Two's Victoria Derbyshire show: "You've got to understand that, of course, there's the European issue but there's also the issue about who is supreme - Parliament or a self-appointed court.
"This is the issue here right now, so I was intrigued that it was a split judgment, I'm disappointed they've decided to tell Parliament how to run its business.
"After all, there was a vote before in December overwhelmingly to trigger Article 50, so they've stepped into new territory where they've actually told Parliament not just that they should do something but actually what they should do and I think that leads further down the road to real constitutional issues about who is supreme in this role."
Mr Duncan Smith said his description of the Supreme Court as "self-appointed" referred to the process of selecting justices with no reference to Parliament.
He told the Press Association: "I do think it is time for us to have hearings for those appointed to the court.
"I think we should have something like the American hearings because we are creating a court which is challenging Parliament's supremacy.
"Parliament should have the final say on the appointments."
Asked if that would politicise the judiciary, he replied: "I think they are politicised already."
Mr Duncan Smith said the courts had previously steered clear of such judgments.
"This is a first. It puts them on a collision course with Parliament."
Labour MP Chuka Umunna said the court's decision allowed the withdrawal from the EU to be triggered by Parliament, rather than "at the whim of the Prime Minister on her own acting as some kind of pseudo-dictator".
The former shadow business secretary said Labour would fight "tooth and nail" to protect the interests of working people during the Brexit negotiation process.
He told the Press Association: "The Prime Minister has served notice that she sees this as an opportunity to change Britain into some giant tax haven serving the interests of the super-rich and multinational companies, and that it is an excuse for her to reduce the social protections that people enjoy at work as consumers.
"We will not have that.
"In terms of the deal that she is negotiating, we will be seeking to make sure that she actually pursues a deal that works for everyone in our country."
Mr Umunna added: "If (Mrs May) can give a speech, she can produce a White Paper setting out in detail what her plan is and what the outline of a deal should be.
"It is not unreasonable for Parliament, acting on behalf of the people, to get her to do that."
Barry Gardiner, shadow secretary of state for international trade, said Labour would examine the forthcoming Bill to see whether it "respects" what the Supreme Court judges said about the importance of Parliament shaping the future.
He told the Press Association: "We will want to see the Government's plan set out very clearly in a White Paper. It simply is not good enough for the Prime Minister to say 'Oh well, I gave a speech last week'.
"A speech is not a White Paper that a select committee or the House of Commons can hold her to account from. A White Paper is a paper committing the Government to pursuing certain proposals. That's what is required."
He added: "What we want to see is that there will be a real opportunity for Parliament to vote at an early stage where that vote can make a difference to the outcome.
"We will not allow the Government to take us into a bargain-basement, offshore tax haven that is a deregulated Britain.
"That is not what people were promised. They were promised at the referendum that Brexit would make people better off, not worse off.
"What now Parliament has the duty and the responsibility to do is to ensure that all of us are better off at the end of this process, and that is what the Supreme Court today has said is the responsibility of Parliament, and not simply the Government alone."