Vacancies in elective county offices shall be filled in the manner and for terms as follows:

   (1)   Sheriff, coroner, register of deeds. In the office of sheriff, coroner or register of deeds, by appointment by the governor for the residue of the unexpired term.

   (3)   County clerk, treasurer, and surveyor. In the office of county clerk, treasurer, or surveyor, by appointment by the county board for the residue of the unexpired term unless a special election is ordered by the county board, in which case the person appointed shall serve until his or her successor is elected and qualified. The county board may, if a vacancy occurs before June 1 in the year preceding expiration of the term of office, order a special election to fill the vacancy. If the county board orders a special election during the period beginning on June 1 and ending on November 30 of any year, the special election shall be held concurrently with the succeeding spring election. If the county board orders a special election during the period beginning on December 1 and ending on May 31 of the succeeding year, the special election shall be held on the Tuesday after the first Monday in November following the date of the order. A person so elected shall serve for the residue of the unexpired term.

   (4m)   Clerk of court. In the office of clerk of circuit court, by appointment of the judge, or by a majority of the judges of the circuit court for the county, for the residue of the unexpired term of the clerk.

   (5)   Supervisors, populous counties. In the office of county supervisor of counties having a population of at least 750,000, according to the last U.S. census, by election for the residue of the unexpired term on the first Tuesday of April next after the vacancy happens, in case it happens no later than December 1 preceding the first Tuesday in April, but if the vacancy happens after December 1 preceding the first Tuesday in April, then such successor shall be elected on the first Tuesday of April of the next ensuing year; but no election to fill a vacancy in the office may be held at the time of holding the regular election for such office. In addition to the elections required under this subsection, the county executive may order a special election to be held under s. 8.50 to fill the vacancy. If an assembly district in the county is altered by legislative redistricting effective prior to the end of an existing supervisor term and a vacancy happens, the person elected to fill that vacancy for the residue of the unexpired term shall be an elector of the assembly district as it existed prior to redistricting.

   (6)   Appointments, how reported. For the information of all concerned appointments by the governor under sub. (1) shall be reported by the appointing officer to the county clerk. Appointments of the county board under sub. (3) shall be reported by the county clerk to the secretary of state. Appointments of clerks of court by a judge of the circuit court under sub. (4m) shall be reported to the county clerk and to the secretary of state.