Most airlines post their fare specials on their sites on Wednesdays.
Remember, though, that these deals are usually short-notice, heavily restricted and often good for long weekends only. This page highlights Web-only deals for travelers who have the freedom to book and travel at short notice. Reservations are made through a toll-free number and no, you don't have to pay with an American Express card. America West's Weekly Specials America West, which has hubs in Phoenix, Las Vegas and Columbus, updates their site every Wednesday with a number of discounted fares. Note that tickets must be purchased by the following Friday at midnight central time.