The Brexit process was thrown into confusion in November when the High Court sided with campaigners who had argued that the government must seek the support of MPs.
Prime Minister Theresa May has said she wants to trigger Article 50 by the end of March 2017, paving the way for formal negotiations that would likely see Britain leaving the EU in 2019.
But with little agreement on what UK-EU relations would look like post-Brexit, discussions on the issue have already proved complex and bad-tempered.
All 11 judges will hear the four-day case -- the first time this has happened since the Supreme Court was established in 2009 -- before giving their ruling in early 2017.
Brexit trigger debate
In the wake of the High Court's ruling in November Gina Miller, one of the claimants in the case, was subjected to death threats and racist abuse, even dubbed "the most hated woman in the country
The three judges in the case were attacked in the pro-Brexit press, with the Daily Mail newspaper branding them "Enemies of the people" in a front page headline that sparked outrage.
In his opening statement, David Neuberger, president of the Supreme Court, spoke out against the "threats of serious violence and unpleasant abuse in emails and other electronic communications" received by participants in the case.
"Threatening and abusing people because they are exercising their fundamental right to go to court undermines the rule of law," he told the court. "Anyone who communicates such threats or abuse should be aware that there are legal powers designed to ensure that access to the courts is available to everybody.
Neuberger said the judges were "aware of the strong feelings associated with the many wider political questions surrounding the United Kingdom's departure from the European Union.
"However ... those wider political questions are not the subject of this appeal. This appeal is concerned with legal issues, and, as judges, our duty is to consider those issues impartially, and to decide the case according to the law."
Leave voters 'betrayed'
British voters backed a "Leave" vote by 52% to 48% in a referendum in June.
More than five months on, there is little clarity over whether leaving the EU would also mean leaving the single market or the customs union.
MPs are unlikely to try and block Brexit, but many are keen to have a say in what form it takes, and the kind of ties the UK would have with Europe afterwards.
The High Court ruling infuriated Leave voters and anti-EU campaigners who said it left them feeling betrayed
, and fearing that Brexit would not go ahead.
"I voted to get out of the EU and there was no suggestion that we'd need to go through parliament to get all the necessary boxes ticked before that happened," Tony Geary from Romford, east London, where 70% of voters backed Brexit, told CNN after the decision.
The court is expected to be packed with campaigners on both sides of the Brexit divide, as well as dozens of journalists; its on-site cafe is preparing to serve up to 600 cups of tea a day.