There is some data available here, enough to put together a chart over time first using the non-adjusted homeownership rates (blue is homeownership, red is the percentage of all homes that are owned without a mortgage):
the effects of wartime rent control and the shift in cash investing to real estate during those years.
Today, the majority of these non-mortgage encumbered homes appear to be owned by senior citizens who have paid off their 30-year mortgages. In light of this, and the growth of the over-65 demographic since the 1940s, it should be somewhat surprising that this statistic has declined.
Here is the same analysis using my age-adjusted homeownership figures (1920 is omitted):
The chart speaks for itself, with a decline in the 1950s and 1960s likely related to a growing use of the 30-year mortgage with its lower down payment requirements, a stabilization through the early 1980s, and a precipitous decline during the cresting and burst of the housing bubble in the late 2000s.
The implied conclusion here, that a dramatic expansion of debt has been necessary just to maintain the illusion of a stable homeownership rate (setting aside the explosion of debt in the 2000s necessary to support an increase in homeownership), puts an even more negative spin on the figures from the preceding post. In short, a decline in homeownership has until the past few years been masked by shifting demographics and an increase in household debt.
One last point here is that although homeownership fell back to its earlier baseline (in non-adjusted figures) following the real estate bubble, equity has not risen back to to the prior 60-70% range, even in spite of the many institutional cash buyers on the market. The debt legacy of the bubble appears as though it will be around for many years to come.