Green’s Dictionary of Slang

(US black) flat, moccasin-like shoes with a design on the toe – often a club, diamond or other card suit – a style of shoe popular among jazz musicians and gamblers.

1977

J.L. Dillard Lex. Black Eng. 91: The shoes, which Morton identifies as ‘St. Louis flats’ and ‘Chicago flats,’ were often specially made for the wearer and were elaborate in design.