Non-refundable airfares are much cheaper than refundable ones, but if you cancel or change your flight, you'll pay a hefty fee. But there are some loopholes and workarounds.
If you are booking an airfare in the United States, U.S. Department of Transportation regulations require that as long as you've booked a non-refundable ticket seven days ahead of your flight, you're entitled to hold your reservation and the fare and change or cancel your reservation within 24 hours of booking, without paying a cancellation fee (typically $200 on the remaining large "network" carriers for a domestic fare, but much more (up to $450 for some international fares), a bit less on other airlines.
You can either cancel the reservation entirely, or change it, within the 24-hour window. If you change it, however, a fare difference may apply, but there is no change penalty. This applies not just to U.S.-based airlines, but any airline selling airfares in the U.S.
You still have to pay for the airfare, and then get a refund without penalty, except that American Airlines is a bit different in that it allows you to hold your seat and the fare for 24 hours without paying for it. On American, you should NOT pay for the fare, but merely choose the 24-hour hold option without payment. If you pay for the fare rather than holding it, you will be hit with a change/cancel fee on American! Also, American sells fare add-ons starting at $68 round-trip that allow you to change your flight for free at any time, and the add-on includes a checked bag round-trip and priority boarding. Something to consider.
Southwest Airlines lets you change or cancel a fare within the 24 hour window without penalty, but it also allows you to change or cancel a reservation anytime before flight time and get a credit for the full amount of your fare, applicable to future travel within a year of the original reservation. You will have to pay any applicable fare increase, however.
Alaska Airlines now allows free changes/cancels if made at least 60 days prior to travel.
Allegiant Airlines is a bit more specific, stating in its rules that you may cancel as long as your scheduled flight is at least 168 hours (24 x 7) away at time of booking.
In order to take advantage of the 24-hour cancel or change rule, it's best to book directly with airlines, either online or by phone, rather than through third-party websites.
And it goes without saying that you can cancel a fully refundable ticket anytime and get a refund, although if you change rather than cancel there may be a fare difference if the fare has changed.
Frequent Flyer Award Tickets, Too?
Does this apply to frequent flyer tickets? I've been able to cancel frequent flyer reservations within 24 hours of booking, and get all fees refunded and miles re-instated without penalty, most recently on British Airways however the DOT rules are unclear on this, and US Airways clearly states that the 24-hour cancel rule does not apply to frequent flyer tickets.
Other Ways to Get a Refund
One more thing: many people don't realize that in airline contracts of carriage, there's a rule (often called Rule 260) about "involuntary refunds." Basically it states that if the airline refuses to carry you for any reason, or if your flight is delayed more than a specified amount of time (121 minutes or greater on AA for example) or the flight is canceled, you can apply for a full refund, even on a non-refundable ticket.
So let's say you buy a fare you no longer can use and the DOT 24-hour rule doesn't apply. You can avoid the change/cancel fee is if your flight is canceled or severely delayed. It may or may not be worth your time to show up for your flight and pray it's canceled or significantly delayed (you do have to check in for the flight).
The Schedule Change Loophole
And you can also get a refund if there's a significant schedule change before your departure (let's say they change you from a 9 a.m. departure to a 6 a.m., or your new flight requires a much longer layover or an overnight stay, or even from a nonstop to a connecting flight). The airline may not notify you of a qualifying schedule change, so if you've purchased a non-refundable fare that you would like to refund, be sure to check the flight schedule to see if it has changed in any way and if it has, call the airline and request a refund, explaining that the schedule no longer works for you (obviously, a change of just a few minutes won't qualify).
Buy Airline Trip Cancellation Insurance
There are so many things that could cause you to have to change your plans, that it pays to buy ticket insurance. Each airline has a different price, and they vary quite a bit from carrier to carrier, so make sure you know the price before you hit submit.