Green’s Dictionary of Slang

1. (US tramp) griddlecakes or pancakes.

1912		Tacoma Times (WA) 2 Nov. 3/5: A ‘string of flats’ is a plate of hot cakes.
1925		J. Stevens ‘Logger Talk’ AS I:3 139/1: He goes forth to eat of [...] a ‘string of flats,’ and ‘larup’ (pancakes and sirup).
1927		‘Jargon of the Und.’ in DN V 446: Flats, Griddle cakes.
1931		Irwin Amer. Tramp and Und. Sl. 75: Flats.– [...] Pancakes, this last another commonly used term in a railroad eating-house, where ‘a string of flats, plenty of pin grease and a tank of murk,’ would be merely an order for a plate of griddle or pancakes with plenty of butter, and a cup of coffee.
1964		(con. 1920s–40s) in J.L. Kornbluh Rebel Voices.

2. (US) small breasts.

1944

E. Wilson 9 Mar. [synd. col.] ‘We tell them to pads on them in the right places and to get their flats fixed [...] But a heaving bosom won’t sell a girl as a singer’.