Google's Ngram Viewer isn't a perfect reference, but it does show that various mentions of "homeless", "homeless person", and "homelessness" exploded in the 1980s.
https://books.google.com/ngrams/graph?content=homeless%2Chom...
I've previously gone through earlier references, and noted that up through the 1960s, references were frequently to persons "made homeless by" some event: a structure fire, a tornado, a hurricane, etc. It was an acute condition, not a chronic one.
https://books.google.com/ngrams/graph?content=*+homeless&yea...
There's also a tie to economic circumstances, if you look at the trace for "homeless man" here:
https://books.google.com/ngrams/graph?content=homeless+perso...
"Transient" was an earlier term:
https://books.google.com/books?id=HC0iAQAAIAAJ&q=%22and+home...
That pairs fairly well with "homeless" in the Ngram viewer, though many of the references are to phenomena other than housing (lunar transits, transition matrices, arterial systems, Markov chains, etc.):
https://books.google.com/ngrams/graph?content=transient%2C%2...
Also "unattached"
https://books.google.com/books?id=vpUbAQAAMAAJ&q=%22and+home...
There were thought to be 50-60k homeless in New York City in 1914:
https://books.google.com/books?id=zAjTAAAAMAAJ&pg=PA29&dq=%2...
Present numbers are 63,000 through September 2018, 130,000 different homeless persons sleeping in municipal shelters over 2017:
http://www.coalitionforthehomeless.org/basic-facts-about-hom...
I've speculated that it might be the financial system itself, and the primacy of the role of financial asset rather than service good of real estate which plays a role -- something I find bordering in lunacy myself, though possibly worth considering:
https://plus.google.com/104092656004159577193/posts/eDscpi6K...