Most expensive months seem to be September, October, late November-Christmas (although the days right after Christmas can be cheaper), then anytime including New Year's Eve. March-May, and June-August can also be higher, the first because weather is nice, and the second because it's the US summer season.

Typically, the most expensive months to fly are December, July, and August. December is driven by the massive demand for holiday travel around Christmas and New Year's, with prices peaking between December 18th and January 2nd. July and August represent the peak of the "Summer Rush" in the Northern Hemisphere, when schools are out and families flock to Europe and coastal destinations. During these months, "Saver" level award tickets and budget fares often vanish months in advance. Conversely, the "shoulder seasons" of May, September, and October offer the best balance of good weather and lower prices. The absolute cheapest months are usually January (after the first week) and February, as demand drops sharply after the holidays and airlines offer deep discounts to fill seats. If you are flying domestically in the U.S., you must also watch out for "holiday spikes" around Thanksgiving in November and Spring Break in March/April, which can temporarily drive prices to summer-like levels.