Ballikinrain School in Stirling, Geilsland Residential School in Ayrshire and the Lord and Lady Polwarth Home for Children in Edinburgh are all Church of Scotland sites being probed by the Scottish Child Abuse Inquiry.
Delivering an update at a preliminary hearing in Edinburgh on Wednesday, Lady Smith said the investigation is "determined to get to the bottom of any systemic failures that occurred".
The probe will also examine historical allegations concerning religious orders including the Benedictines, Sisters of Nazareth and the Christian Brothers, as well as their relationships with the Catholic Church in Scotland.
Panel member Glenn Houston said: "We know that the scale of the harm caused to children in some institutions was systemic, i.e. it happened frequently, not just to one child but to many, it was sometimes perpetrated by more than one employee and it was sustained because of failures such as poor management oversight and inadequate processes of supervision, investigation and reporting to the relevant authorities."
The Daughters of Charity of St Vincent de Paul, the Sisters of our Lady of Charity of the Good Shepherd, the De la Salle Brothers and Marist Brothers are also the focus of investigations under the inquiry.
Lady Smith went on to say: "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."
Charged with providing recommendations to improve legislation, regulations and practises in Scotland, the inquiry was established in October 2015 and is due to report to ministers within four years.
The inquiry will also investigate how children were treated by care providers including Quarriers and Barnardo's, and schools including Loretto School in East Lothian and Merchiston Castle School in Edinburgh.