Court of Public Opinion". On 6th May, the people were finally permitted their say in the General Election and their "verdict" will be subjected to analysis for the days and months to come.
As widely forecast, the result was a "Hung Parliament" and the recently issued Cabinet Manual Guidance clicked into operation. This guidance is not based on rules set out in a formal constitution. It is based on a number of historical precedents such as Edward Heath in 1974. It permits the incumbent Prime Minister to remain in place even in a situation in which his Party does not have the largest number of seats in the House of Commons and even when his Party did not capture the largest share of the vote.
There are a number of possibilities.  the Conservatives and Liberal Democrats combine in some way. This could be either (a) a coalition arrangement or (b) a "Confidence and Supply" arrangement. The question of electoral reform is likely to be a major obstacle to this possibility.  the Labour Party and Liberal Democrats combine in some way in order to exclude the possibility of a Conservative government. This has been described as a "Coalition of Losers" but some commentators have seen it as a more likely outcome in policy terms than  though, due to personalities (Clegg v Brown), it may prove to be impracticable. A third possibility is that the Conservatives try to form a minority government but that would be unlikely to offer the political stability which the country requires in the present state of international finance etc. This could not come about without the resignation of the Prime Minister (e.g. if he is unable to achieve possibility 2 and form a government able to command the confidence of the House of Commons) whereupon Her Majesty could ask David Cameron to form a government. The fascination of all this is that possibilities 1, 2 and 3 may not be the only ones since there are some 29 MPs from other parties and they may be in a strong position to make demands as a price for supporting whatever government emerges.
The Cabinet Office is known to have made facilities available to assist with discussions but an important feature is that Her Majesty is not involved in any of the negotiations. . That is essential in a "constitutional monarchy" in which the Crown has a limited role in the formation of "Her Majesty's Government".
Addendum 1: Election Outcome:
Seats: Conservative 305; Labour 258; Liberal Democrats 57 and Others 29
Votes: Conservative 10.68 million; Labour 8.6 million, Lib Dem 6.81 million; Others 3.51 million
% of vote: Conservative 36.1%; Labour 29%; Lib Dem 23%; Others 11.9%
Addendum 2: An undemocratic Possibility?
There has been a suggestion that Labour might "ditch" Gordon Brown and form a (minority) government with Lib. Dem. support. Surely, such an outcome would surely be profoundly undemocratic. It is one thing to argue that, according to the Cabinet Guidance, an incumbent P.M. can remain whilst matters are sorted out. It is quite another to argue that the incumbent P.M. could resign but his Party continue under a new leader as the major player in a power-sharing minority government. After all, Labour have been decisively rejected (Scotland excepted) in the polls.
Addendum 3 (10th May): Gordon Brown stated that he would be resigning once a new Labour leader was elected. This would appear to make the option of a Labour/Lib Dem arrangement more likely.
Addendum 4 (11th May): Talks between the Liberal Democrats and the Labour Party took place but there was no agreement. Interestingly, a number of senior Labour Party figures had said that such a deal would lack legitimacy - see Politicshome. Consequently, Gordon Brown resigned as Prime Minister and then H.M. The Queen requested David Campbell to form a government. The twists and turns of the period 6th May to 11th May 2010 and the various personalities involved will interest political commentators for a long time to come. See, for example, "Big Beasts turn on Mandelson and Campbell" and "Labour infighting killed coalition hopes".