Shinplasters are small-denomination banknotes or forms of paper money. The face value of shinplasters is usually less than $1. The name probably arose from similarities to plasters used on sore shins.
The term goes back at least to the early 19th century, when some banknotes in the United States were issued for as little as 12.5 cents. In 1834, Senator Thomas Hart Benton, a famous “hard currency” advocate who referred to himself by the nickname “Old Bullion,” was heard to say, “What! Do you want a coroner’s jury to sit and say, ‘Old Bullion died of shinplasters?’” (Schlesinger, 1945).
Adam Smith in the Wealth of Nations does not mention shinplasters per se but devotes a good deal of attention to discouraging the issuance of small denomination notes. According to Smith:
Where the issuing of bank notes for such very small sums is allowed and commonly practiced, many mean people are both enabled and encouraged to become bankers. A person whose promissory note for five pounds, or even for twenty shillings, would be rejected by everybody, will get it received without scruple when it is issued for so small a sum as a sixpence. . . . It were better, perhaps, that no bank notes were issued in any part of the kingdom for a smaller sum than five pounds. Paper money would then confine itself . . . to circulation between the different dealers [wholesalers]. . . . Where paper money is . . . confined to circulation between dealers and dealers, as at London, there is always plenty of gold and silver. Where it extends itself to a considerable part of the circulation between dealers and consumers, as in Scotland, and still more in North America, it banishes gold and silver almost entirely from the country. (Smith, 1952, 139)
In 1829, Great Britain banished banknotes of less than £5. In the United States, a treasury circular of April 1835 disallowed the acceptance of notes under $5 for payment of federal obligations. In February 1836, the treasury extended the ban to notes under $10, and prohibited banks holding U.S. government deposits from issuing such notes. In April 1836, Congress enacted legislation banning notes under $20.
In April 1862, the Congress authorized the issuance of a paper currency in fractional units of a dollar. At first, the currency took the form of postal stamp designs engraved on notes, but later took other forms. This fractional currency could be exchanged for legal-tender notes in amounts up to $5. In 1875, Congress provided for the replacement of fractional currency with subsidiary coinage.
In 1870, a shortage of silver coins led the Dominion Government of Canada to issue paper currencies in denominations of a fraction of a dollar, familiarly known as shinplasters because of their small size. The Canadian government discontinued the shinplasters in 1935 after issuing over 300 different varieties of shinplasters over the 65-year period. 
The fact that history is inflationary may have put an end to shinplasters. The United States has attempted to replace the $1 bill with a coin, suggesting that the dollar bill is beginning to fall into the range of small change, in the same value range as the shinplasters.
See also: Inconvertible Paper Standard, Postage Stamps
Angus, Ian. 1975. Paper Money. 

Chown, John F. 1994. A History of Money. Schlesinger, Arthur M. J. 1945. The Age of Jackson.
Smith, Adam. 1776/1952. An Inquiry into the Nature and Causes of the Wealth of Nations.