This is a question that often crops up. The Council has limited powers to do so and any power is contained with the Dog Control Act 1996. For this posting, I will only deal with the aftermath of your dog being seized for an alleged attack [section 57 & 58 Dog Control Act 1996].
If your dog is seized after an alleged offence [for example attacking another dog or person] then Council cannot [generally] euthanize your dog [there is a provision if the dog is so violent that it cannot be safely seized and there is a threat to the safety of council staff or members of the public, however, this does not often happen and Council must be able to evidence their actions]. The council must hold the dog until a decision has been made to prosecute you or not. If a decision has been made to prosecute then you can apply for the release of your dog pending the outcome of the prosecution. When you do this it must be in writing and the fees applicable for seizure and sustenance must be paid.
If the council decide not to release your dog they must issue with a section 71 [Dog Control Act] Notice of Retention. This decision can be appealed to the District Court. Whilst awaiting the outcome of the prosecution the council cannot euthanise your dog. Only two ways exist for that to take place  you authorise the council to do so and surrender your dog to the ownership of the council in writing for that purpose [there is a form] and  when the court orders the destruction of your dog. That is it, there is no other power to do so. [again noting I am not dealing with a roaming dog that has been impounded and not collected [that will be a later post].
I am aware of a practice whereby the dog owner is told that the sustenance costs will be thousands of dollars when a case is finished at court, this often puts fear into the dog owner and they surrender the dog. I have a concern with this practice as it is often put to the dog owner when they are upset and vulnerable. There is no doubt if you do not have your dog destroyed there will be a cost down the line. However, case law does not support the Council being able to obtain those costs via a court order after you have been convicted, it is not within that Courts power. Ultimately the Council will have to seek the debt via other means. Again seeking early advice is essential. The initial call is free so do not wait.