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1st fiat system (France)
John law, a Scottish gambler and amateur economist, was able to convince to the regent that the main reason for the economic to slow down is due to gold and silver were too scarce and inelastic to serve as money.
 [Source: www.mapforum.com]
He proposed by switching to paper, trade would be faster as more currency is created. He set up a bank that took deposits in coin, but issued loans and withdrawals in paper.
His newly created Banque Royale (Royal Bank) issued 2.7 billion livres in banknotes in the space of two years. His newly created Mississippi Company achieved a market capitalization of 5 billion livres over the same period. This resulted in a massive stock market bubble. Following a parabolic blow off, the bubble collapsed, the bank failed and Law fled the country, leaving destitution in his wake. (Antoine Murphy, John Law (Oxford, 1997), passim.)
 [Source: Frontispiece of Arlequin Actionist (Amsterdam, 1720), reprinted in Antoine Murphy, John Law (Oxford, 1997)]
However, people wanted gold and silver when they took profits. Law capped redemption in gold and silver to avoid depleting his reserves. This removed France's paper currency from the gold and silver standard and hence put it on the Mississippi Company share price standard. The amount of paper currency afloat was now many times the actual reserves of gold and silver and hyperinflation set in.
In 1720, the bank and company were united and Law was appointed Controller General of Finances to attract capital. Law's pioneering note-issuing bank was successful until the French government was forced to admit that the number of paper notes being issued by the Banque Royale were not equal to the amount of metal coinage it held. (The French Period" (of New Orleans area), 2009)
2nd fiat system (United State)
The second episode was the case of the Continental Congress. This is the most sympathetic of the four. Meeting in May 1775 following the outbreak of hostilities at Lexington and Concord, Congress had a war to finance and no clear way to do so.
 [Source: www.americanrevolution.org]
It could not levy taxes because its authority was unclear. Both the British Crown and the individual states claimed the power to tax, and anyway, a brand new tax would not have gone down well in the context of a rebellion that was largely about taxes.
A loan was also out of the question, since a lender would have been crazy to take the risk of funding a ragtag band of colonial rebels at the outset of their rebellion. So Congress did what it had to do, and printed up the money.
Lots of it.
  [Source: www.frbsf.org]
Over the next five years, until they stopped the presses in 1780, Congress issued about $241 million face amount of irredeemable, non-convertible paper bills known as “Continentals.”11 The bills served their purpose, keeping the armies in the field, but how they functioned in practice is described in the following passage (William G. Anderson, The Price of Liberty (University Press of Virginia, 1983), p. 3.):
A barber wallpapered his shop with Continentals. An old soldier, wounded in the leg, used a bundle of his pay as a bandage, and coined the word “shinplaster,” which was later used to describe any sort of money that could not be redeemed. A ship’s crew discharged in Boston, and paid off in now worthless currency, found a way of making suits out of the paper bills and paraded through the streets. “For two or three years we constantly saw and were informed of creditors running away from their debtors, and the debtors pursuing them in triumph, and paying them without mercy,” wrote [a contemporary observer].
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