A range of data sets at local authority level including household estimates, housing market and house prices, rents, lettings and tenancies and homelessness are provided at gov.uk (external link).
Also provided is:
Data regarding rough sleepers (external link) at a local level.
Data regarding Gypsies and Travellers (external link) at a local level.
Data regarding housing tenure (external link)
The English Housing Survey (external link) is a continuous national survey that collects information about people's housing circumstances and the condition and energy efficiency of housing in England.