Here are The Traveling Professor's tips on how to get a flight upgrade.
The coveted flight upgrade is not easy to get. It is nearly impossible to get a flight upgrade from coach to another class of service (Premium Economy, Business or First Class) if not part of an airline loyalty program like American Airlines AAdvantage program or Delta Airlines SkyMiles. Even if a flight is overbooked or there is some sort of service disruption requiring the re-booking of a flight or the changing of a seat, those with elite status through a loyalty program are most likely to get those upgraded seats before those who are not in a loyalty program.
Upgrades on domestic flghts are easier to get. In fact, most airlines will automatically or by request, upgrade their customers on domestic flights if there is availability. This is done up to 3 or 4 days before departure. Otherwise they can be requested and purchased by using loyalty points or miles.
The Traveling Professor has flown well over one million miles internationally and is a member of an airline loyalty program. He has never received a free upgrade on an international flight. Any upgrade he has received has been by using one of the strategies outlined below.
One way to get upgraded on an international flight is to redeem frequent flyer miles or award points. Sometimes a cash payment is also required. For example, on a coach class flight from New York to Paris on American Airlines, a passenger on a discount fare will be required to use 25,000 miles plus pay $350. In fact, the entire American Airlines upgrade chart can be found here. It is important to note that upgrades are given based upon the number of seats airlines have allocated for upgrades. It is better to request the upgrade as far in advance as possible, or at the last minute.
Bid for Upgrades: The Professor flies on IcelandAir once or twice a year. If a flight is not fulled to capacity, sometimes Iceland Air has asked me to "bid" against other passengers bid on availble upgraded seats. I can recall paying about $600 for a coach seat on IcelandAir and winning a bid on an upgrade to their "Saga Class" for about $300. To purchase the Saga Class seat outright would have cost about $1500. Here is a list of airlines that allow bidding on upgrades and will tell how to bid on a flight upgrade.
**UPDATE** See this article from "The Points Guy" with tips regarding upgrades on American Airlines.