Since deregulation in 1978 US airlines have been free to charge whatever they want, and they almost universally responded with a plethora of extraordinarily complicated fare rules and an almost infinite number of fare options designed to maximize their revenue in a highly competitive marketplace. As a result literally every time you search for an airfare you get a different number.
Before 1978, US airfares were approved by regulators and published, both in the airline's own timetables and in consolidated publications like the Official Airline Guide (OAG).
It might be possible to research an old airfare by finding old timetables or copies of the OAG, which are sporatically available online. For example according to this old American Airlines timetable from February 10, 1930, the fare from Chicago to Peoria was $6.71.
The Bing flight search engine stores historical fare information which it will display for you when you search for a flight. After you search for a particular city pair you'll see a link marked "Details & fare history" which, sometimes, shows you some historical data for airfares on that route. When this is available, it will give you approximate information as to whether the current fare offered on that route is historically high or low.