The remarks come after Mayor Jules Radich suggested, in the wake of singer Ed Sheeran not coming to Dunedin, the city should launch a "Dunedin Sounds" festival.
Mr Radich suggested Dunedin Sounds could be an event Sheeran might one day headline along with other international artists, including Taylor Swift.
Cr Steve Walker took to social media to criticise Mr Radich’s thinking.
"It’s a reminder of what a complete buffoon this guy is, he’s literally rambling on like a baby Trump.
"He did this pre the last local election and enough of you fell for it."
Cr Brent Weatherall said Cr Walker’s "nasty style" on social media did him no favours.
"He would be better to invest in a pair of six-inch platforms if he wants to be noticed, and stick to issues important to himself if he wants to be taken seriously by voters opposed to disrespectful personal attacks on the mayor."
Cr Bill Acklin said Cr Walker’s behaviour was unacceptable and he accused the Labour-endorsed councillor of being disruptive through the term.
"I find Cr Walker to be an embarrassment to the governance of this city," Cr Acklin said.
"I would like to think that the party who endorse him would also be ashamed of his unprofessional antics."
Southern Labour electorate committee chairwoman Ruth Chapman said the party continued to have confidence in Cr Walker as a candidate for a council seat.
Cr Walker stood by his position.
"In essence, [Mr Radich] can’t just bleat out any nonsense about Taylor Swift et al, with no robust background work to understand how mechanically it would all work.
"There’s never been a vision or masterplan under Radich, just plucked-out-of-the-air nonsense.
"Jules isn’t a bad or nasty guy — he just isn’t capable of the job voters picked him to do."
Mr Radich did not want to engage with Cr Walker’s comments.
"Cr Walker persists in name calling and such behaviour says a lot more about him than his targets."
Mayoral candidate Andrew Simms said Cr Walker’s comments crossed a line.
"I have seen this and many other similar comments from Cr Walker which communicate a deep disrespect for the current mayor and a deep-seated resentment of previous decisions made by Mayor Radich.
"While Cr Walker may not respect Mayor Radich as a person, he should have respect for the office of the mayor.
"It is entirely possible to express disapproval in the performance of the current mayor and uphold some dignity and respect at the same time.
"In my view, such exchanges only serve to further undermine the community’s respect for this current council at every level."
Cr Carmen Houlahan, who is also running for mayor, said Cr Walker’s comments were in the atmosphere of an election campaign.
"We are in an election. Have you seen some of the abuse we are getting?
"Sadly, ‘buffoon’ is kind compared to some of the unacceptable comments from some in the public."
Cr Houlahan said criticism of the council as dysfunctional was wide of the mark.
"This term we have worked hard with a huge workload, with government changes to legislation around Three Waters, the hospital campaign ... Most of the time we have all worked well together.
"We have differing views at times — that is democracy."
Cr David Benson-Pope said: "My view of the mayor appeared in your publication as the front-page lead when I stated ‘he has none of the skills required for the job’."
— Additional reporting Grant Miller
