A bid to oust Tony Blair as party leader would require a challenger to have the backing of 20% of Labour MPs.
Mr Blair won the Labour leadership in 1994
That translates to 71 out of the 355 MPs elected on Thursday.
If a would-be candidate secures this level of support they must then write to Labour's general secretary announcing their intention to run.
The contest would then be decided at conference by an electoral college of unions, MPs and constituency parties.
Normally the party leader and his deputy are formally re-elected at Labour's autumn conference, whether there is a challenge or not.
But in the event of a leadership bid, constituency parties and unions would be expected to ballot their members on which candidate they favour.
The electoral college is split three equal ways between Labour MPs, party members and members of affiliated trade unions, on a one member one vote basis.
The rules for triggering a contest are different if the leader resigns or is incapacitated.
Mr Smith took over as Labour leader from Neil Kinnock
If the leader is prime minister at the time then an election can only proceed if requested by a majority in a card vote at conference.
When in government, the Cabinet can appoint an interim party leader, but this does not necessarily have to be the deputy leader.
if a vacancy occurs and a leadership election is called, any candidate needs to secure the backing of 12.5% of the Parliamentary Labour Party, or 51 MPs, in order to put his or her name forward.
In 1994, Mr Blair won the Labour leadership following a contest triggered by the death of John Smith.
He saw off challenges from acting leader Margaret Beckett, now environment secretary, and John Prescott who became deputy leader and then deputy prime minister.