I\'m in the trade of developing business web applications. They are mostly internal-use, sometimes with a partner zone, but rarely available to the public internet. They are
It can be done, and be effective. However, it is usually not being done well. There is a tendency to be seduced by the dark side of Flex, and forget about the basics. The result is a plethora of truly horrendous applications that break expected user behavior.
Of course this is not limited to just Flash/Flex. I was brought in as a consultant on a website targeted at iPhones. Because of their implementation of CIUI, the back button and bookmarking functionality of the web browser is broken. "Oh we thought about that already". They had created an elaborate way to store your own page favorites on their server, assuming you are logged in of course. And a series of links insured that the user would never need their back button. I am not exaggerating here either, I kid you not.
Failures at this basic level are what you want to watch out for. Very nice applications can be built using Flash and Flex, and can solve browser compatibility issues. Just be careful to not forget the fundamentals.
It's already happening? Flex!
Many people and organisations are doing this with both Flex and regular Flash.
It's worth noting that my bank uses Flash for its login screen. They're doing some sort of biometric typing rhythm identification that I think would be a security hole if it was in Javascript.
The downside is that it means I can't use their damn site on my mobile phone!