At the Lifestyles Centre, people can share their own memories, Monday to Thursday from 8am to 9pm and on Friday from 8am to 6pm and at the Memorial Hall, Monday to Friday, from 9am to 4pm.
Provost Margaret Cooper was the first to sign the book of condolence in Hamilton, along with council leader Joe Fagan.
She said: “It was very humbling to be the first person to sign the book and, although my message was personal, I also signed on behalf of the people of South Lanarkshire and for anyone who, for any reason, cannot sign one of the books themselves.”
Those who can’t make into the towns can sign an online condolence book at https://www.royal.uk/send-message-condolence.
Following the death of Her Majesty The Queen, flags across the country are currently flying at half mast. In South Lanarkshire, this includes the flags outside the council’s headquarters in Hamilton.
There was a temporary change as part of national protocols, established with Buckingham Palace.
Those protocols called for flags to be restored to full mast for the Proclamation of the Accession of the new king at St James Palace in London on Saturday at 11am, followed by a number of local proclamations on Sunday.
As a result, flags were flown at full mast from 11am on Saturday to 1pm on Sunday, then lowered again to half mast where they will remain until after the period of national mourning.