What we now call a brassiere. In the south a brassiere is still referred to as a tit sling.
Wallace Reyburn wrote a book describing the lawsuit over rights to the bra, fought from 1934 to 1938 in New York, between a German-born designer, Otto Titzling (1884-1942), and a French-born designer, Philippe de Brassiere.
Stripper or slut-in-a-box worker. Mostly you hear the expression from old gomers nowadays. From the old usage of "sling" to mean delivering or presenting, so in 30s hep jive, waitresses were hash slingers, bartenders were booze slingers, and so on.
Grandma got through the first part of the Depression slinging milk, but the dairy went bankrupt, so she slung hash for a while, till she found out she could make a lot more at the burly-q as a tit slinger.