Sacida Cali, of Blackthorn Court, Hall Road, Stratford; Faiza Sharif Hassan, of Leyton Green Road, Leyton, and Fahmmo Abdi, of Stockfield Road, Walthamstow, all suffer from health problems as well as overcrowing and applied to the London Borough of Waltham Forest to be rehoused during 2004.
The families have been told they face a wait of five to ten years before they can be rehoused under the scheme.
But the three mothers say the system is too clumsy to fairly assess families like theirs with multiple needs and today asked Mr Justice Lloyd Jones, sitting at London's High Court, to declare the scheme unlawful.
Under the scheme available housing is advertised by the council fortnightly and applicants for council housing bid for homes they wish to live in.
Applicants are ranked as having either no preference, reasonable preference or additional preference, the latter being the highest rating.
The house is then allocated to the bidder with the highest ranking who has been waiting the longest.
But Kevin Gannon, for the three mothers, argued the scheme is unlawful because it is 'too vague and unstructured' and doesn't reflect the complicated needs of families such as those before the court today.
He told the court that Miss Cali lives in a two-bedroom flat with six children aged between six and 18, three of whom suffered from chronic medical conditions.
Miss Abdi lives in a one-bedroom flat with four children and her niece, and has suffered from mental illness exacerbated by the overcrowding.
Miss Hassan lives in two rooms with two children and suffers from a kidney condition which is made worse by having to share a bed with her daughter.
It is complex combinations of needs such as those of the three mothers that the council's scheme is too blunt an instrument to cope with, Mr Gannon argued.
'How can the scheme comply with the statutory requirements? Because of the inconsistency and arbitrariness which the scheme lends itself to, it is not a system which any reasonable authority could legally adopt.
'The scheme contains no mechanism for producing a fair assessment of applicants with multiple needs,' he argued.
'The council's allocation and choice-based lettings policy is unlawful and the decisions made by them in the case of each of the claimants is unlawful, if the scheme they flow from is unlawful,' Mr Gannon concluded.
Ranjit Bhose, for the Council, admitted that the system 'may not be perfect' but argued that it is by no means unlawful and merely follows Government guidelines on council housing allocation.
'There may well be anomalies in such an approach and the scheme may not be perfect, but it is an acceptable scheme,' he said.
'All that the council has done is move towards a more simplified banding system consistent with the Government's preferred approach to allocations,' Miss Bhose concluded.
Recognising the importance of the case, Mr Justice Lloyd Jones reserved his judgement on the women's judicial review challenge until a later, unspecified, date.
STRAND NEWS SERVICE