Singh was one of India's longest-serving prime ministers and he was considered the architect of key liberalising economic reforms, as premier from 2004-2014 and before that as finance minister.
Among those who paid tribute to Singh on Thursday were Prime Minister Narendra Modi, who wrote on social media that "India mourns the loss of one of its most distinguished leaders".
Modi said that Singh's "wisdom and humility were always visible" during their interactions and that he had "made extensive efforts to improve people's lives" during his time as prime minister.
Priyanka Gandhi, the daughter of former prime minister Rajiv Gandhi and a Congress party member, said that Singh was "genuinely egalitarian, wise, strong-willed and courageous until the end".
Singh was the first Indian leader since Jawaharlal Nehru to be re-elected after serving a full first term, and the first Sikh to hold the country's top post. He made a public apology in parliament for the 1984 riots in which some 3,000 Sikhs were killed.