There are several recent children's books on the theme "The Cat Sat on the Mat". It is also quite surprising how often the phrase is used in other contexts, for example in stylistics.
The origin of the phrase is undoubtedly quite remote. The oldest quote I have found dates from 1933 in the first edition of Edith Sitwell's book "The English Eccentrics". On page 243, it reads:
‘Turkish. Eighteenth century.’ The pages were few; and after an interval of respect we inquired: ‘What is it about?’ Mr Waley, with sudden animation: ‘The Cat and the Bat. The Cat sat on the Mat. The Cat ate the Rat.’ ‘Oh, it is a child’s book.’...
(Public domain, at least in Canada, United States, etc.)
Dame Edith Louisa Sitwell, Order of the British Empire, (7 September 1887 – 9 December 1964) was a British poet and critic and the eldest of the three literary Sitwells. She reacted badly to her eccentric, unloving parents and lived much of her life with her governess. She never married but became passionately attached to Russian painter Pavel Tchelitchew, and her home was always open to London's poetic circle, to whom she was generous and helpful.
Sitwell published poetry continuously from 1913, some of it abstract and set to music. With her dramatic style and exotic costumes, she was sometimes labelled a poseur, but her work was praised for its solid technique and painstaking craftsmanship. She was a recipient of the Benson Medal. (Wikipedia)