Singh, president from 1982 to 1987, had been in hospital in a serious condition since a car accident in late November. Authorities in the northern state of Punjab have launched an investigation into the crash.
The government declared seven days of official mourning for India's only Sikh head of state. His body will be cremated in New Delhi today.
Singh held the largely ceremonial post of president while Sikhs were embroiled in one of the country's greatest upheavals since independence. His selection, orchestrated by then-prime minister Indira Gandhi, was seen as a move to curb Sikh militancy in Punjab, where most of India's Sikhs live.
He faced a crisis in June 1984 after troops stormed the Sikhs' holiest shrine, the Golden Temple in Amritsar, to flush out militant Sikhs demanding independence for Punjab. But his official role forced him to remain a passive bystander when the troops stormed the temple.
Many Sikhs felt Singh should have resigned and he was denounced by their five high priests, who forgave him only after he played a crucial role in negotiating the withdrawal of troops from the shrine.
Mrs Gandhi was shot dead by two Sikh bodyguards four months later, and on the advice of the ruling Congress Party, Singh named her son Rajiv to succeed her.
But he fell out with the new prime minister. Singh, who once said he would sweep floors for Mrs Gandhi, publicly accused Rajiv of failing to keep him informed on important issues.
The accusation was an embarrassment to the prime minister and while the controversy flared it threatened to enmesh India in a constitutional crisis. The dispute between the two men had cooled by the time Rajiv Gandhi was assassinated by a suicide bomber in May 1991.
Singh began his political career in the 1930s, leading a campaign against the princes who ruled his native Farikdot district of Punjab. He was jailed in solitary confinement from 1938 until 1943 for political activities.
Singh joined Mrs Gandhi's cabinet as home minister from 1980 to 1982.