Various elements of the modern supermarket arose in different places, mostly in the 1920s.

The first self-service grocery store is thought to have been a Memphis Piggly Wiggly opened in 1916. The concept was patented and franchised to other operators. Customers entered through a turnstile and made their way through the store to a checkout at the exit.

Drive-in markets that patrons reached by auto began to be built in Southern California about 1923. These were akin to farmers markets (a strong regional tradition) in hosting a variety of independently owned stands, but sellers of packaged goods also set up shops. The low prices and convenience began to set the pattern of weekly shopping excursions by auto—to a single place.

In the 1920s, Ralphs began to build branch stores in new outlying areas of Los Angeles, and within a few years the branches were every bit the equal of the downtown store. At first, the new Ralphs were organized like other groceries or department stores, with goods called for and presented at departmental counters. But Ralphs displayed produce in ways that made customer inspection easy, and within a few years introduced self-service in other departments, assumed ownership of all the departments in the store, and eventually created a modern "front end" of multiple checkstands in one row, allowing one payment at one time for all of a customer's purchases.

Two Houston food markets, J. Weingarten and Henke & Pillot, began to evolve into similar prototypes about the same time, introducing large parking lots that the stores faced regardless of how that oriented the store's front in regard to traditional retail streets or streetcar lines. Both these markets and Ralphs began to think of their stores as freestanding objects in the landscape, rather than storefronts on an established shopping street.

These innovations spread rapidly in the early years of the Depression, with King Kullen's in the New York area focusing on major brands at rock-bottom prices, with large stores (some repurposed from industrial spaces) where products were displayed in stacks that, to customers of the time, seemed warehouse-like. Soon it was clear that this new way of food marketing was not a short-lived fad related to the Depression, and established chains like A&P or Safeway began to follow the new model. In 1937, Piggly Wiggly introduced a larger wheeled shopping basket in an Oklahoma City store, and that came to be the iconic representation for the supermarket concept.

I've drawn extensively from Richard Longstreth's 1999 book The Drive-In, the Supermarket, and the Transformation of Commercial Space in Los Angeles, 1914-1941.