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In reaction to the two recent crashes of Boeing 737 MAX aircraft, a number of countries and airlines have stopped flying this plane while the latest incident is investigated.

I imagine this would put enormous stress on the rest of the fleets of these airlines. Aircraft are expensive, it's hard to see how they could have enough spares of comparable aircraft that they can find substitutes for all of these flights.

Do airlines generally have contingency plans for something like this? Or are there likely to be lots of flight cancellations as a result?

The FAA hasn't taken this action (yet). Apparently this jet is not yet a major component of US fleets. CNN says that of 548 that have been ordered by US airlines only 65 have been delivered so far, and if they were grounded it wouldn't cause major disruption.

But I suspect there would be a bigger impact in some smaller countries.

BTW, I was unsure whether to post this on Travel or Aviation SE. Moderators, feel free to migrate it if you think it's more appropriate on Aviation.

There are no airlines that fly only this airplane. So for example, Air Canada has cancelled all its flights from Halifax to London, which use that plane. It's re-routing passengers through Ottawa, Montreal, or Toronto, where the flights use different planes. Some people will cancel, since they didn't allow for this kind of re-routing delay, and the rest will be absorbed into significantly fuller planes for a few days.

One of the options for affected airlines is to lease aircraft as an interim measure.

There are quite a few companies that have aircraft available to fill in as replacements. They operate either on a "wet lease" basis (including crew) or on a "dry lease" basis (without crew). You can even find an A380 if you want.

Availability varies a lot based on the time of the year, so they could have planes available now, while in the summer it may be a lot more difficult (as many of those places are used for charter flights during that period).

Higher-end airlines on long-haul routes usually do not like to use those much, as they are often second-hand planes, and their interior equipment may be quite different from what they usually provide to their customers (often more geared towards the charter business than business travel), but if that helps avoid cancelling dozens of flights, it's better than nothing.

Airlines do not make money easily, and losses are very common due to many reasons, crashes or bans are one of them. Hence their plans usually include some scenarios like this. I am sure that their ERP teams (emergency response teams) are dispatched as we speak to solve many problems, including media, recovery plans, etc.

The most logical and feasible solution in case their fleet cannot cover the demand for a short period ban would be some sort of an agreement with another airline to handle the passengers until things get sorted out. This can be achieved quickly and does not require a long term contract and will not result in loosing customers as they would be able to fulfill the great promise of moving them from point A to B safely and on time.

Wet or dry lease (as mentioned in another answer) require a lot of preparations, including certification and training, it's not like hiring a car, it's a very long process and usually the contracts are long enough which makes it not the best option for this case which I assume it won't take long.

FYI, many of IATA training courses which are taken by almost all airlines include scenarios like this. Each airline has an ERP team from different departments that gets activated in such cases to do their pre-defined roles. Their main goal is to get out of such situations with the least possible damage in terms of reputation (the most important impact for long term) and to find options and solutions that would minimize the short term financial impact, such as the aforementioned temporary agreements.

There are 2 impacts, one if financial.

It costs money to ground planes (and do full inspection and certification again?) and replace them with other existing planes; not just physically have different planes, but juggling the different range that those planes have and put them on appropriate routes.

For example, Air Canada re-route passengers from to Montreal or Toronto to board different planes that can fly out to the UK (for example)

"In a statement, Air Canada says it is rebooking affected passengers through Montreal, Toronto and Ottawa. Affected customers are asked to contact Air Canada Reservations to change their flights free of charge."

https://globalnews.ca/news/5047559/boeing-737-max-8-ban-halifax/

The other impact is legal.

If it is found that there is a flaw in the plane system (hardware, software) or in the training related to new system, then a large civil suit can (and will probably) happen.