Congress President Sonia Gandhi said the verdict heralds a new era for individual rights, personal liberty and human dignity.
Slamming the Modi government, she said the judgment strikes a blow on the "unbridled encroachment and surveillance by the state and its agencies in the life of the common man".
Congress Vice President Rahul Gandhi said the decision was "a major blow to fascist forces" and the rejection of BJP's ideology of "suppression through surveillance".
The Modi government has drawn much flak for making Aadhaar mandatory for availing several services including banking services and purchasing a SIM card.
Former Union Minister and senior lawyer P. Chidambaram said that when the Congress-led UPA government conceived Aadhaar, it was meant to be "an administrative tool" to ensure the benefits of government schemes reach only the right people.
Communist Party of India Marxist (CPI-M) leader Sitaram Yechury said the ruling will have far-reaching consequences in various domains.
"A far-reaching judgment which will have consequences in various domains, as technology is playing a greater role in our day-to-day lives," Mr Yechury said, adding that CPI-M has opposed making Aadhaar mandatory that could result in data misuse by foreign technology firms.
West Bengal Chief Minister Mamata Banerjee too welcomed the Supreme Court's ruling.