According to the Denver Post this past Sunday (infographic, page 5E), there are 12,050 homeless people in Colorado, of whom about a third live in Denver. This includes 600 "chronically homeless" in Denver (and presumably on the order of 1,200 elsewhere in the state). Chronically homeless is a euphemism for vagrant and defined as "single people who sleep outside and in shelers."
The graphic reports (seemingly with regard to the homeless as a whole, although I suspect that the reference might actually be to vagrants), that 25% of mentally ill, 40% are alcohol or substantive abusers and 15% are both (which I suspect means that the total number of mentally ill, alcohol or substance abusers is 50% not 80%, although the phrasing is ambiguous).
Something like 95% of the homeless are basically down on their luck -- people who have lost a job, can afford a place to live, have had a family break up, often single mothers or families with children, a substantial minority working.
According to a related story, Denver's efforts have brought down homlessness 11% overall and reduced "chronic homelessness" by 36%. Boulder has also seen drops. The city says it saves $1000-$4000 a year (excluding federal and Medicaid savings) by finding people homes. Panhandling on the 16th Street mall is allegedly down 92%.
Together with statistics from our prison population, homelessness illustrates the high price we pay for poor behavioral health services (most notably deinstitutionalization a generation ago), and the price we pay for having a weak safety net.
The numbers also suggest that vagrancy is not an intractible problem. The vast majority of the homeless do not fall into vagrancy, and there is good reason to believe, given the concentrated mental health and substance abuse problems behind vagrancy, that helping the couple thousand who are in this state will not cause others to fall into the same trap. This costs money. Vagrants, almost by definition, have serious problems. But it isn't insanely expensive and the experience of "housing first" efforts has been that people allowed to get on their feet and given the help that they need can do so. The trouble is worth the effort, for those in that dire state and for the rest of us.