Homes for All Act of 2021 This bill provides funding for the construction of new public housing units, authorizes additional assistance for the construction of new affordable housing units, and otherwise addresses housing affordability. Specifically, the bill provides permanent, mandatory funding for the capital and operating funds available to public housing agencies (PHAs) and allows PHAs to use such funds to construct new public housing units. Under current law, a PHA generally may not use such funds to construct public housing units if doing so would result in a net increase of the number of public housing units owned, assisted, or operated by the PHA. The bill also (1) provides annual funding through FY2032 for the construction of new public housing units; and (2) authorizes additional assistance through FY2032 from the Housing Trust Fund for the construction of new, private-market, affordable housing units. A PHA may not discriminate, with respect to the rental of a unit developed with assistance under the bill, based on an individual's criminal record or immigration status. Additionally, a PHA receiving such assistance must meet specified requirements with respect to the new units regarding other forms of discrimination, supportive services, environmental standards, and accessibility. Finally, the bill requires the Department of Housing and Urban Development to establish a Community Control and Anti-Displacement Fund to award grants to local governments for the purpose of combatting gentrification and neighborhood destabilization.