Modified from an earlier answer

Homelessness in the USSR is an interesting topic because it exposes a number of other social problems and systemic dysfunction within the Soviet state apparatus. Like other modern industrial societies, there was no single overarching cause for homelessness, but there were specific aspects of the Soviet milieu that exacerbated this problem among its population.

For one thing, the Soviet Union was an incredibly vast and heterogeneous economic and geographic entity. This made it difficult for the state to impose its model of a proletarian industrial state that provided full employment and a high quality of life. Although the Soviet state was able to eliminate a great deal of extreme poverty as a whole, not all areas of the USSR were developed equally. This was in evidence on one of the persistent problems of the Soviet state: housing. The tendency of the state to prioritize gigantic industrial concerns coupled with wartime destruction meant the large Soviet cities seldom could adequately house their population of workers. Although the Khrushchev era alleviated the housing shortages greatly through the use of prefabricated rebar-concrete structures, these buildings could still be quite cramped and unsatisfactory for family living. Furthermore, maintenance for these buildings could be somewhat patchwork and this became an issue during the low years of the Brezhnev-era economic stagnation.

Field-research on Soviet homelessness of the 1970s and 1980s found that stresses within the family helped fuel the Soviet homeless problem. While this particular cause for homelessness is far from unique, there were specific aspects of Soviet family life that could make the problem of an unhappy family worse. The twin historical crises of both Stalinism and the Second World War added strains to some Soviet families as children lost one or both parents. For war orphans, Soviet orphanages and group homes were frequently underfunded and their wards subject to various abuses. Remarriage could also potentially introduce new strains in family life. This led to both incidents of juvenile delinquency and runaways. The Soviet police and good deal of the public saw this as a crisis of youth hooliganism, especially in the 1950s, and Soviet youth charged with these offensives often found themselves sent to work camps or other reformatories. For a lot of youthful offenders, they became a marginalized underclass later in life. The labor colonies and youth hostels were not surprisingly quite harsh and the state was more concerned with observing this population than providing for it. In a state that regulated both movement and residency, the official stigma of a criminal record made it very difficult for individuals to break out this system in adulthood. These problems in the family, socially-charged policing, and anemic social safety net helped further encourage transiency.

There were other aspects of the Soviet state and society that enabled Soviet homelessness. Unlike youth vagrancy, the state tended to ignore alcoholism as social problem and this had a ripple effect through Soviet society. Not only could alcoholism contribute to stresses in the family, but drunkenness created problems with violence and in the workplace. Severe alcoholics became pariahs within large parts of Soviet society and police forces often linked vagrancy with alcoholism. The Soviet health system was ill-prepared to deal with alcoholism, which made treatment difficult. On a related note, Soviet mental health care was notoriously deficient throughout the existence of the USSR and Soviet psychiatry was quite a different animal than in the West. Soviet psychiatry tended to identify mental health problems as fundamentally biological in origin. Soviet discourse on mental health focused heavily on issues of "abnormal minds," (in the words of Khrushchev), and treated those with mental health issues as if there was something physically wrong with them. This meant that those who suffered from problems of mental health frequently did not get effective treatment, but instead suffered social ostracization and exclusion. This perception of mental abnormality extended to the Soviet discourses on vagrancy. One 1984 Soviet study of the problem framed vagrancy in harsh physiological/psychological terms:

spending the nights at train stations, at boiler rooms , in lofts and in other places unsuitable for living, negatively affects the mental state of the vagrants and as a result they lose the sense of physical and psychological discomfort and lose the desire to stop this way of life.

Within this context, Soviet individuals who found themselves vagrants for whatever reason faced a series of stark alternatives. When caught, the state often tried to force them to relocate to group work camps or dormitories within the Soviet periphery were they could be observed. The quality of life at these facilities left much to be desired and many elected to escape. The other option was to carve out a space in the underground and grey areas of the Soviet economy. While nominally free of state regulation (although the danger was always there), this meant interacting with a hardened criminal element. In both options, these individuals suffered from social death and were not considered either by the state or society at large to belong to the Soviet experiment, but rather were often put among the scapegoats for its failures.

Sources

Eaton, Katherine Bliss. Daily Life in the Soviet Union. Westport, Conn: Greenwood Press, 2004.

Feldbrugge, F. J. M., Gerard Pieter van den Berg, and William B. Simons. Encyclopedia of Soviet Law. Dordrecht: M. Nijhoff Publishers, 1985.

Hagenloh, Paul. Stalin's Police: Public Order and Mass Repression in the USSR, 1926-1941. Washington, D.C.: Woodrow Wilson Center Press, 2009.

LaPierre, Brian. Hooligans in Khrushchev's Russia: Defining, Policing, and Producing Deviance During the Thaw. Madison: University of Wisconsin Press, 2012.

Stephenson, Svetlana. Crossing the Line: Vagrancy, Homelessness, and Social Displacement in Russia. Aldershot, England: Ashgate, 2006.

Looks like this has been asked before:

https://www.reddit.com/r/AskHistorians/comments/2tb7er/how_prevalent_was_homelessnesspoverty_in_the/

Are there no documented articles left outlining specific social programs?

Yes, there were homeless people in the USSR, but it's virtually impossible to know how many because it was illegal to be homeless as well as unemployed. Also everyone had to register at his/her address, and tell the government where do you live (and if you won't — it's also a crime). This was called propiska (прописка). Propiska was highly regulated (for example, it was very hard to get propiska in Moscow, if you were not born here, and if you don't have propiska, your chances to get a real job are, like, zero). And if you are not homeless, but you are living at the wrong address — there will be problems with law.

Homeless people were called 'bomzh(i)' — "without a defined residence".

So, what are you options if you are homeless. The first one is jail time. I don't think there are any open stats about that, but it was not very popular solution, because even then it was not perceived as real crime. The second option is working camps/dormitories, especially in rural areas (with lack of working hands). Homeless guys were used as a cheap labour with some hope that this process will help them to become real and useful citizens. The third option is just to be homeless as long as you can (if you are lucky or unlucky, I don't know).

As for the movies. If you want to see the real USSR, I would highly recommend 'Office affair' with subtitles (https://www.youtube.com/watch?v=cmylNwnR9qQ and https://www.youtube.com/watch?v=JPjr65ncLL8), 'Moscow doesn't believe in tears' (Oscar-nominated, and it's about 1950s and 1970s at the same time) or 'Garage' (quite depressing, crazy and funny). All these movie are works of fiction, and obviously they are somehow polished, and soviet life was shown a little better than it was, but I believe you will get the better understanding of people lives/hopes/fiers from them than from any documentary.

These are comedies and melodramas, so you won't heard anything about Gulag here, but Gulag or Stalin's 'chistka' or fear were not parts of everyday life (in 1970-1980s, at least). At the same time these movies are not about how it is good to live in the USSR ('Garage', for example, is a 2-hour drama/comedy about places on the parking lot, and how evil could be people to each other just because they want to have a place for their car), it's about ordinary people.

Source: Poverty in Russia: Public Policy and Private Responses, edited by Jeni Klugman

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