Labour did pilot and plan a Bedroom tax for social housing
In Hansard you can find this
Mr. Clifton-Brown: To ask the Secretary of State for Work and Pensions when the local housing allowance will be extended from the nine Pathfinder authorities to all local authorities. [R] 
Malcolm Wicks: There will be a comprehensive evaluation of local housing allowance Pathfinders including interviews with claimants, landlords, local authority staff and stakeholder organisations, as well as an analysis of administrative data. This is already underway.
The findings of the evaluation will inform decisions on the national extension of the allowance. It is envisaged that local housing allowance will be extended to all local authorities in 2006.
Mr. Clifton-Brown: To ask the Secretary of State for Work and Pensions for what reasons the local housing allowance applies only to the de-regulated private sector. [R] 
Malcolm Wicks: We hope to implement a flat rate housing benefit system in the social sector, similar to that anticipated in the private rented sector to enable people in that sector to benefit from the choice and flexibility that the reforms can provide. We aim to extend our reforms to the social rented sector as soon as rent restructuring and increased choice have created an improved market.
Mr. Clifton-Brown: To ask the Secretary of State for Work and Pensions (1) how the rent of those tenants whose rent is higher than the local housing allowance will be paid; [R] 
(2) whether there will be a hardship payment to prevent the eviction of those tenants whose rent is higher than the new local housing allowance. [R] 
Malcolm Wicks: Tenants whose rent is higher than their local housing allowance will be expected to make good the difference with their landlord. This is no different to what happens under existing rules. During the Pathfinder stage, no claimant will be worse off financially at the point of change as they will be covered by a form of transitional protection.
There will not be a hardship payment to tenants whose rent is higher than their local housing allowance. Tenants will have the choice to shop around and look for a cheaper property in such circumstances.
Labour did bring in limits on spare rooms for people in private rented property in 2008. It is clear from this that their plan was to later on introduce this for people in Council Houses and other Social Housing. The delay was because they wanted the rents to raise to be close to those in the private sector. "as soon as rent restructuring and increased choice have created an improved market."