For this you look to state law. Utah State law contains two general area of protection for some municipal employees. Member of fire departments and police departments in cities of the first and second class are members of the classified civil service and are subject to dismissal from their jobs only for things like misconduct and incompetence. Even when fired for one of the permissible reasons these employees have the right of due process, which generally must consist of pre-termination notice of the reasons for dismissal and a fair hearing. This section provides job protection for non civil service members who are not department head, or superintendents.
Some employees clearly have no statutory job protection.
They include city managers, police and fire chiefs, and heads of departments and most appointed officers. Some appointed officers have a specific term and cannot be removed except for misfeasance or malfeasance in office. An example of this is a justice court judge. Some appointed officers in third class city and towns such as the recorder and treasurer are appointed by the mayor with the advice and consent of the council and cannot be terminated except with the cooperation of the council. In third class cities and towns appointed officers hold their offices until their successors are appointed and qualified.