They are among more than 100 locations where the abuse of children is alleged to have taken place, Lady Smith announced.
Boarding schools, institutions run by religious orders and local authorities are among the establishments being probed.
The schools specifically being investigated by inquiry staff are Fettes College, Gordonstoun, the former Keil School, Loretto School, Merchiston Castle School, and Morrison’s Academy when it was a boarding school, the hearing was told.
Faith-based organisations being looked at include those run by religious orders including the Benedictines, Sisters of Nazareth and the Christian Brothers.
The inquiry is also investigating the relationship between the Catholic Church in Scotland and these religious orders, as well as three establishments run by the Church of Scotland.
Inquiry chair Lady Smith named a list of places being probed as she provided an update on its progress during a preliminary hearing in Edinburgh.
The Scottish Child Abuse Inquiry is examining historical allegations of the abuse of children in care and has been taking statements from witnesses since the spring of last year.
Lady Smith told the hearing at Parliament House that the inquiry is “determined to get to the bottom of any systemic failures that occurred”.
She appealed to anyone with relevant evidence to come forward and speak to the inquiry.
“We are determined to find out the truth about what happened to children in care, where, how and why,” she said.
“We want to find out why the abuse was not prevented, why it was not stopped, and what needs to be done to protect children in care in the future.”