The association was commenting following the move by the Western Isles Council to advertise a third time for the rector's post at its largest secondary, the Nicolson Institute in Stornoway.
The most forceful response came from Michael O'Neill, director in North Lanarkshire, who rejected the heads' claims. Of four secondary appointments made or about to be filled in his authority, there had been more than 20 applicants for each school, all of high quality and all with the Scottish Qualification for Headship.
Mr O'Neill said he would not expect a huge number of applicants since the pool is confined to deputes or heads in smaller secondaries. But he felt this could improve since there is now an expanded group of deputes to choose from with the disappearance of assistant head posts. "So I'm not at all pessimistic about the future," he said.
The picture in other large authorities suggests the problem may be in primaries rather than secondaries. George Gardner, depute director in Glasgow, said the city had around 20 applicants for each of the two secondary vacancies in the past seven months.
"In the primary sector, however, we have had to readvertise on several occasions. Sometimes we have been down to four or five applicants," Mr Gardner said. Smaller primaries were less attractive than larger schools, particularly where there may be a question mark over their future.
Mr Gardner hoped that the development of the council's strategy for pre-12 campuses would alleviate the problem because these would be larger schools, with a management structure including deputes and principals.
"At present, many primary heads feel isolated where they may be the only person holding a management post," he said.
But in Edinburgh, Colin Dalrymple, depute director, said there were fewer candidates generally for heads' posts in both primary and secondary. His council faced a particular difficulty in filling Catholic primary headships. "There was virtually no interest in the last couple of cases," Mr Dalrymple said.
In addition to the changing nature of the headteacher's job and the demands facing management, Mr Dalrymple felt Edinburgh may have an additional problem in attracting candidates from outside the city because of the high cost of housing.
Stirling also reports little interest in primary headships in small, remote schools such as Crianlarich and Inversnaid; they had, respectively, five and two applicants last year but decided to make no permanent appointments and acting heads were put in charge. The vacancy in St Mary's primary in Bannockburn attracted three applicants and one candidate was selected.
Fife, on the other hand, reveals a "reasonably healthy" interest in the top school jobs, between 10 and 12 for three secondary headships in the past year and from nine to 16 applying for eight primary posts.
The council says the vast majority of candidates come from within Fife, but vacancies at large secondary schools do not seem to attract existing heads.
Garry Crosbie, a senior manager in the education department, noted that internal candidates seem to be in a stronger position if they have already been acting heads. Possession of the SQH does not always guarantee success, he said.
Bruce Robertson, director in Highland, has an "in between" message. While recent adverts for vacancies at secondaries in Alness and Kingussie attracted 13 and nine applicants respectively, the authority faces real difficulties in primaries. "Both large and smaller schools have had to be readvertised and this is one of the reasons that we are extending the cluster headship pilots," Mr Robertson said.
"Generally, I think the big issue is mobility of applicants as well as a decision on life-work balance especially for primary applicants."
The most whole-hearted agreement with the HAS came from Leslie Manson, Orkney's director, who referred to "a huge problem recruiting at all levels outside the central belt". The last three or four headships, Mr Manson said, attracted very few candidates.
Orkney has begun to think more radically, moving to linking headships and using short-term secondments out of school to target up and coming staff.