What problem do MOM (Message Oriented Middleware) solve? Scalability? Integration?
In which domain are they typically used and in which domains are they typically no
To address your specific questions:
In which domain are they typically used and in which domains are they typically not used?
Like databases, messaging systems crop up everywhere.
For example, say, is Google using such solution for it's main search engine or to power GMail?
Google uses a lot of home grown technology, but a lot of their open source contributions and known use cases suggest that messaging is (or should be) central to some of the main services.
What about big websites like Walmart, eBay, FedEx (pretty much a Java shop) and buy.com (pretty much an MS shop)? Does MOM solve a need there?
Very much so.
An example use case is scaling web page requests. When the user makes a web request, the web server puts it onto a queue for background processing. This means that the web server can keep working while the request is processed. It also means that the web server does not need to know how the request is handled, making system maintenance, upgrade and rollback much simpler because the main parts are 'decoupled'.
So, anyway, the web request gets processed by a back end service, or possibly by many services, eg 'look up book titles', 'draw shopping cart', 'get advertisement', 'check user account'... Finally all the results get put onto another queue, ready for collection and user response by the web server. Typically the system will include a timeout of around 100ms so that any late requests just get thrown away. The user sees anything that got processed in the time interval. This is one reason why some large ecommerce sites have pages that appear to load in stages.
There are many more use cases...
Does it make any sense when you're writing a Webapp where you control the server-side and have an homogenous environment (say tens of Amazon EC2 instances all running Linux + Java JVMs) there and where the clients are, well, Web browsers?
Definitely. If you have an unknown, or unbounded, number of users, server side instances, and application latencies, then it makes sense to use messaging, even if just as a scalable substrate for non-blocking RPC.
Does it make sense for desktop apps that need to communicate with a server?
In lots of cases. One very common case is when the server pushes events to the desktop app, eg game event, tweets, price feeds in finance, system alerts....
Or is it 'only' for big enterprise stuff where you typically have a happy mix of countless of different systems that needs to communicate in a way or another?
Definitely not only for those 'legacy integration' cases but they are important too. At RabbitMQ, the biggest customers we have in terms of pure scale or message volume are cloud providers and big web application providers.