One way of reading the result isthat there is at least a majority of Members of the Scottish Parliament who favour holding a further referendum on Scottish independence, The SNP and the Green Party campaigned on pro-independence manifestos - SNP manifesto - Green manifesto. Therefore, the likelihood is that political pressure will now increase for a further independence referendum. It is a pressure which may prove difficult for the UK government to resist.
Analysis of the Scottish election may be seen at Financial Times 8 May 2021, BBC News- Election 2021: The fourt key takeaways in maps and charts and BBC News - Professor Sir John Curtice - What the 2021 election results mean for the parties
The new Scottish Parliament will elect a Presiding Officer. It is an impartial role and whoever is elected will give up their Party membership. The Scotsman 10 May looks at possible candidates. The Presiding Officer does not normally vote, but has a deciding vote if there is a tie. If the vote is for Stage 1 of a Bill, the Presiding Officer will usually vote to allow the Bill to continue to Stage 2. At other times, the Presiding Officer will usually vote against a motion or amendment.
How could a further referendum lawfully come about?
The UK government could agree to a referendum being held and ensure that a so-called "Section 30 Order" was made in order to give the Scottish Parliament competence to legislate for the referendum. This was the mechanism used in 2014 (previous post) when voters rejected independence (55% to 45%). Such agreement to hold a further independence referendum cannot be ruled out but, at the time of writing, is perhaps unlikely.
If the UK government were to refuse the Section 30 route then an interesting question might arise - Does the Scottish Parliament have powers under the Scotland Acts to legislate for a referendum in Scotland to determine whether the Scottish electorate wishes Scotland to become an independent nation.
The Scotland Acts:
The Scotland Act 1998 created the Scottish Parliament and defined its powers. Certain matters were reserved to the UK Parliament at Westminster.
Section 29(1) of the 1998 Act provides that - An Act of the Scottish Parliament is not law so far as any provision of the Act is outside the legislative competence of the Parliament.
Section 29(2) goes on to provide that - A provision is outside that competence so far as any of the following paragraphs apply -
(a) - (not relevant here)
(b) it relates to reserved matters,
(c) to (e) - not relevant here.
Section 29(3) - For the purposes of this section, the question whether a provision of an Act of the Scottish Parliament relates to a reserved matter is to be determined, subject to subsection (4), by reference to the purpose of the provision, having regard (among other things) to its effect in all the circumstances.
Scottish Parliament legislation to hold a referendum would appear to relate to a reserved matter - i.e. the Union of Scotland and England.
I find it hard to read the words "relates to" in any other way but argument could possibly be put forward to the effect that the legislation will do no more than test public opinion in Scotland on the question of independence and that the legislation is of an "advisory" nature not requiring politicians to act on the outcome. At the time of writing, we do not know precisely what the arguments might be.
Political resolution of this issue should be the preferred way but the question may eventually end up before the Supreme Court.
Draft Independence Referendum Bill (March 2021):
In March 2021 the Scottish Government published its thinking as well as the text of a draft bill - Draft Independence Referendum Bill.
Relevant case law:
Martin  UKSC 10
Imperial Tobacco  UKSC 61
Continuity Bill Reference  UKSC 64
Links and Developments:
The New European 13 May 2021 - Seven reasons why Scotland should stay in the Union
BBC 9 May 2021 - Scottish Independence: Could the Supreme Court rule on an independence referendum?
S. Tierney, ‘The Scottish Parliamentary Elections and the “Second Referendum” Debate’, U.K. Const. L. Blog (10th May 2021) (available at https://ukconstitutionallaw.org/))
Cabinet Office - Dunlop Review 24 March 2021
House of Lords Constitution Committee on the Future Governance of the UK
House of Lords Constitution Committee 25 May 2016 - Union and devolution
House of Lords Constitution Committee 18 March 2015 - Intergovernmental Relations,
10 May 2021.