The fall of Morarji Desai's government signals the start of a long period of political instability for India as leaders there try to patch together a new national coalition from a hodge-podge of regional parties, according to diplomats and State Department sources.

Analysts here think it is unlikely that any new government will be able to hold together very long without a complete realignment of the party structure.

"It's the same thing that is going on in Italy," said one diplomat here who keeps a close watch on the internal politics of India.

The ruling Janata government is a fragile coalition of six disparate political parties that was stitched together in 1977 to provide a unified opposition to Indira Gandhi. While the Janata has run the country for the past 28 months, it has never built a truly national base.

Moreover, its members have always kept one foot in their own political parties, and it was this inability to work together that caused the fall of the government Sunday.

The jockeying among Indian political leaders began in earnest yesterday, one day after Desai's sudden resignation as the fourth prime minister of the world's most populous democracy.

Charan Singh, the 77-year-old finance minister who spent the last week working behind the scenes to topple Desai's government, opened his bid to take over by resigning from both the government and the Janata coalition.

Singh claimed the support of Gandhi, the former prime minister who remains India's best known political figure even though she is given little chance to take over the government at this time.

This is an abrupt about face for Singh, who was a severe critic of Gandhi and has led the fight to punish her for the excesses of the emergency rule that brought about her defeat in 1977. He and Desai both were jailed by her during the emergency.

Since his resignation, Desai has continued as head of a caretaker government and of the Janata coalition, and has pressed for first crack at forming the new government.

Political instability, he said, "threatens the very foundation of democratic government."

But Chandra Shekhar, the Janata president, called on Desai to step down as the leader of the coalition so others could form a new government.

One of those waiting for Desai to bow out of the Janata leadership was Defense Minister Jagjivan Ram, who with Singh is given the best chance to become the new prime minister. Ram, the leader of India's 100 million untouchables, has refused to make his bid to form the government as long as Desai, 83, is still in the running.

Singh's power base among the farmers with small landholdings, who are influential in northern India, puts him in direct opposition to Ram, represents the country's poorest class.

It will be up to India's president, Neekam Sanjiva Reddy, to decide who shall have the chance to set up a new government. Reddy, the first politician to hold the largely ceremonial post of president, appears to be taking an active rol in trying to forge a new coalition to rule the country.

Despite the rivalries among them, India's politicians appear to agree that they do not want to call a new election at this time. According to analysts here, none of the parties have built up the kind of national organization they need to wage an election campaign. The next election is scheduled for 1982.

Despite defections, Desai's Janata is the largest single bloc - in the lower house of the Parliament, holding about 200 of the 538 seats. Singh claims to have 87 votes, but he is counting on getting more from other parties in the Parliament and from further defections from the Janata, which lost the votes of almost 100 members last week. CAPTION: Picture, Charan Singh, right, who helped topple India's prime minister, talks to reporters as an aide signals victory. UPI