In particular as not to know shit from Shinola, the phrase not to know something from Shinola means to be completely unaware or innocent (Shinola is also used with lower-case initial)—cf. also not to know — from a bar of soap.

It refers to Shinola, the trade name of an American brand of shoe polish. The earliest mention of this name that I have found is from the following advertisement published in the Miners’ Journal (Pottsville, Pennsylvania) of Saturday 23^rd November 1901:

The “Coulter” Shoeholder holds the shoe in a firm position while off the foot and makes shoe cleaning and shining easy. Adjustable to any size and removable when not in use. Price $1.00 including polishing cloth and polish. The “DANDY” SHINER is a good one, too, at the same price. SHINOLA and BLACKOLA are the leading blackings for men’s and ladies’ shoes. Half soles from 10 cents a pair up.

C. A. SEIDEL.
728 WEST MARKET STREET.

The following image is from trade-marks registered September 8, 1903, in the Official Gazette of the United States Patent Office (Government Printing Office – Washington, D.C., 1904):

41.082. Shoe-polish. American Chemical Manufacturing & Mining Co., Rochester, N. Y. Filed Dec. 8, 1902

Shinola
The word “Shinola.” Used since January 1, 1900.

(Shinola is from shine and -ola, a suffix used to form nouns denoting commercial products—cf. for example granola, which appeared in 1886 with capital initial as a proprietary name for a breakfast cereal devised by W. K. Kellogg.)

The earliest instance that I have found of the phrase not to know something from shinola is from The Des Moines Register (Des Moines, Iowa) of Sunday 16^th June 1946—sugar is probably euphemistic for shit:

“I don’t know sugar from shinola when it comes to fighters,” said Carl Hubbell, the once great screwball artist at Toot Shor’s the other night, where a group of writers was discussing the coming heavyweight bout.
“But since everyone seems to be voicing an opinion I’ll make a prediction.”

In the second-earliest instance of the phrase that I have found, the contrast is also between sugar and shinola, but the sentence is affirmative; it is from an article about “the state of dilapidation into which both Republican and Democratic political organizations have fallen in California”, published in The Newhall Signal (Newhall, California) of Thursday 24^th July 1947:

Any political observer that knows sugar from shinola knows that the root cause of party disorganization and confusion in California is the California political code.

On Thursday 18^th May 1950, the same newspaper, The Newhall Signal (Newhall, California), published an article titled Sample ballots show identical candidate lists, in which the phrase is affirmative and the opposition is between ice cream and shinola:

Every politician who knows ice cream from shinola knows that cross filing gives him a Chinaman’s chance of copping both nominations in the primary and thus save himself the expense of another campaign in November.

The word used in contrast to Shinola is sometimes chosen according to the context; for example, in an article titled Grapevine Mob Not Mugged: Winter Tough, Robins Are, Too; Form Gang, by Kenneth Nichols, published in the Akron Beacon Journal (Akron, Ohio) of Saturday 3^rd February 1951:

It was the Avon st. call that did it. A woman reported the mob was eating the seeds off her grapevine or something. “Lew,” called the city editor, summoning a largish, rugged photographer named Lew Henderson.
“Go out to Avon st.,” the editor ordered. “Get me a picture of a flock of robins eating a grapevine—raw.”
Mr. Henderson did not change expression. He is used to odd orders for pictures. And apparently he did not know robins from Shinola.

The earliest instance that I have found of not to know from Shinola (that is to say, with elision of the word used in contrast to Shinola) is from The Austin Statesman (Austin, Texas) of Thursday 15 May 1952, in which a boxer named James D. Turner was reported as explaining why he had retired from the ring:

“There are too many guys running the show who don’t even know what’s going on, managers who don’t know from Shinola and fighters who let the managers push ’em around.”

The earliest known instance of not to know shit from Shinola is from the following limerick, dating from 1948, published in the chapter Excrement of The New Limerick: 2750 Unpublished Examples, American and British (Bell Publishing Company – New York, 1977), edited by Gershon Legman (1917-99), American cultural critic and folklorist:

There was an old man from Arcola
Who didn’t know shit from Shinola.
     He pined and he pined,
     For his shoes were unshined
When hernia stopped up his hole-a.

In Lo, the Former Egyptian! (Doubleday & Company, Inc. – New York, 1947), the American journalist and humorist Harry Allen Wolfgang Smith (1907-76) used a periphrasis for shit:

Indiana is a fascinating state in spite of the sorry beginnings it had. It used to be inhabited by tow savage peoples—the Indians and the whites. The Indians didn’t have much sense and were inclined to eat each other. It took a long time to bounce them out of the state, but the white settlers who swarmed in from the East and South managed to do it by giving them whisky until they were unable to distinguish between sheep droppings and Shinola.