They would have set out the details of the process under which councillors could "call in" some contentious decisions for reconsideration.
Minister Mark H Durkan said councils will be "left to their own devices" to determine processes to protect minorities, as a result.
He said he was "disappointed" that the DUP had blocked the regulations.
Mr Durkan,  the minister in charge of local government reform, said he would have to go "back to the drawing board" to try to get consensus.
He said that the new councils had been "left without a clear direction" on the issue less than six weeks before they took on their powers.
He said that as the principles of the "call-in" process and of qualified majority voting had already been included in legislation, councils would be "left to their own devices" as to how to operate the mechanisms.
Under the "call-in" procedure, a council would have to reconsider a decision if 15% of councillors express concern that it would disproportionately affect a section of the community in an adverse way.
An 80% majority would then be required for the decision to be voted through.
The DUP tabled a petition of concern, which meant that the regulations did not pass, in spite of all parties except the DUP voting for them.
DUP MLA Peter Weir said the party was concerned that the regulations would mean "the most vexatious application in the world" for a call-in would be successful.
He said that there should be a provision for legal opinion to be taken into account on applications for a call-in.
Mr Weir also said that the party had no issue with the "general thrust" of having a call-in process or qualified majority voting, but believed the assembly had to get the details right.
The DUP has blocked regulations for new councils in the assembly.