I've got a use of "call shenanigans" that dates to 1998, and I strongly suspect that, even if it isn't the earliest use, it's the source of the phrase's popularity over the last decade or so.
In "Cow Days," the thirteenth episode of the second season of South Park, the boys are at a carnival playing games which they come to suspect are rigged. Kyle calls shenanigans, which brings Officer Barbrady to investigate. However, the carny allows the boys to win the game when the officer is present, and Kyle retracts his call of shenanigans.
Later in the episode, the boys discover that the prizes they were trying to win are in fact cheap knock-offs, and Kyle formally reinstates his claim of shenanigans. The claim is upheld this time by Officer Barbrady, which gives the townspeople free rein to destroy the carnival with brooms.
Again, this may not be the very first use of the term, but you asked whether anyone had made it particularly popular, and I think this is what did it. I saw the episode when it was new, and I remember quite distinctly thinking that I'd never heard the word shenanigans used that way before—but within days of the airing of that episode, several of my friends were calling shenanigans every chance they got.
I believe this simply derives from one sense of the word fat:
fat 2a well filled out : of sizable proportions : THICK <a ~ letter> <a ~ volume of verse> : BIG <a resistor spark plug ... permits a wider gap, thus a fatter hotter spark — Newsweek> : unusually large <he had to pay a ~ price to move his factory — Martin Turnell> [Websters 3rd New Int'l Dictionary]
The term is used ironically. At face value it means there is a large chance of something happening, but underneath it really means there is a slim chance after all.
Best Answer
The OED's first quotation for successfully "[passing] with flying colours" is:
Partridge's A Dictionary of Cliches (1978) says it is "Originally of a man-of-war."
Colours has been used for a flag or ensign flown by a military ship since at least Shakespeare's time. The first three quotations in the OED:
And the OED has "with flying colours" for a regiment or ship flying a flag, ensign or standard:
The phrase is sometimes also "to come off with flying colours", and the OED's 6th sense of the verb come, first recorded by Shakespeare, is:
So to come off with flying colours is from a battleship being able to successfully leave the battle, undefeated, with its flags and ensigns still flying proudly.
Antedatings?
The OED has the literal "with flying colours" from 1612 (adj. n.). I found some possible antedatings from 1609, but they're "with their colours flying" (n. v.) in A Generall Historie of the Netherlands by Edward Grimeston. Page 120:
Page 501:
Page 790:
Page 1316: