The latest meeting was called following a report by the council’s top legal officer, Alex Bougatef, who said it was his legal duty to declare the local authority is breaking the law.
Presenting the report, Bougatef said the lack of anyone in charge raised serious concerns about decision-making, accountability, and effective governance.
He said: “The appointment of a leader is not optional. It’s a statutory requirement.”
On Thursday evening, seven proposals were put forward and all of them failed to pass, the Local Democracy Reporting Service said.
Labour and Reform put forward bids to form a minority administration, as well as an alliance made up of smaller parties and an independent councillor.
Reform also put forward proposals that would see councillors elected at the ballot box in May 2027, arguing it should be up to the people to decide.
However, other parties argued this was not a “quick fix” and described the proposal of further elections as “passing the buck”, warning the council would still require a leader over the coming nine months before any election.