went to press, most of the other authorities said they were still checking as part of wider nationwide safety reviews called for by the Scottish government in the wake of last week's tragedy at Liberton High.
Twelve-year-old Keane Wallis-Bennett was getting changed for a physical education class at the school when a free-standing wall beside the showers fell on top of her on Tuesday last week.
Inverclyde Council - one of the authorities unsure if its schools had any modesty walls - said the structures were not inherently hazardous. "It should be stressed that this kind of wall is not dangerous if properly constructed and maintained," a spokesman said.
But the lack of detailed information more than a week after the schoolgirl's death has brought fresh calls for action to improve student safety.
The National Parent Forum for Scotland warned that local authorities should know exactly how many of these walls were in schools and their condition before students returned next term.
Chairman Iain Ellis said: "I would expect that, by the end of the Easter holidays, all councils would have inspected all their schools and identified any issues. I would say that is the deadline, and that is what parents would expect."
But Ken Cunningham, general secretary of School Leaders Scotland, defended local authorities that did not know how many modesty walls existed in their schools. "I am not surprised councils don't know the answer to that question. They would never have thought about it until this happened," he said.
Mr Cunningham predicted that current concern could lead to other issues being detected. "It will raise the question in headteachers' minds, `Is there anything else in the school that we have concerns about, that others have voiced concerns about, or that we have concerns about which have not been raised?'," he said.
Larry Flanagan, general secretary of the EIS teaching union, sought to allay fears over the walls, which were not part of the design of modern schools. "By and large, schools are relatively safe places. Local authorities are quite mindful of health and safety," he said.
None of the councils surveyed by TESS blamed budget cuts for jeopardising school safety, and several said that funding in that area had actually increased rather than fallen.
But the responses did little to reassure critics who were outraged over reports that Liberton High was not placed in the highest-priority category for repairs by the City of Edinburgh Council because of financial pressures.
More than 200 schools and leisure and community centres have now been inspected in the capital in an attempt to locate and check all free-standing walls. One, in a bicycle shed at Craiglockhart Primary School, has since been pulled down.
Edinburgh council said that further demolition could take place after Easter, with nine similar walls found in changing rooms and toilet facilities at Castlebrae Community High and Leith Academy. Smaller walls have also been found in 11 primary schools. Some of these support sinks where students wash their hands and have been cordoned off as a precaution.
Earlier this year, Edinburgh council was fined pound;8,000 for breaching health and safety regulations during an incident at Liberton High in 2011, when a schoolgirl was seriously injured after she fell more than five metres down a lift shaft.
An investigation into the latest incident at the school by Police Scotland and the Health and Safety Executive is ongoing.
Ian Murray, Labour MSP for Edinburgh South, said: "These checks need to happen as quickly as possible, although it may be a mammoth task, depending on the size of school estates.
"It will be beneficial for parents to know that, by the time their children have come back, councils have checked all schools to make sure any of these walls are safe."
Liam McArthur, Liberal Democrat education spokesman and MSP for Orkney, said: "The fact that so few councils are aware of whether their own school estates include walls of the kind that collapsed last week is a real concern. Parents will want to be reassured that local authorities have a grip on school safety. We need to see this issue gets the attention it deserves right up to Cabinet level."