I have the feeling that Flash-based ( or Silverlight-based) websites are generally frowned upon, except when you are creating games or multimedia-content rich applications.
"I was about to click checkout but I wanted to change something so I hit the back button"
Flash isn't evil, it's what people do with it that's evil.
For goofy vector animations and maybe even for example a 360 product view or interactive diagram, yes. Flash can do beautiful Full HD animation on my ancient box @ 60 fps using ~15 % cpu.
For web video, if you have no other choice, I suppose.
But for entire websites and these things called 'RIA's, no.
So shockingly a technology works better for the thing that it was designed to do.
In its defence, most of the issues raised here about how people have implemented Flash in their websites, not about Flash itself. Flash does support accessibility it's just that most people don't consider it when building their sites. Flash does work on mobile phones - it's Flash Lite, although the ActionScript is limited in earlier versions.
Why should anyone have an automatic right to view the source code of a website? A web author has spent time and effort dreaming up their code to share their ideas through the medium of the web. If you want to know how something works, why not put a bit of effort in yourself, and work it out? The beauty of the web is the message, not the medium.
So what if Flash is owned by a single company - Adobe bought it for a reason, and that's because it's a fantastic bit of kit. The problem comes from web authors not using it properly, trying to make it do things it was never intended to, or simply not applying standards to their sites when developing using Flash.
What is so evil about a Flash based website? Absolutely nothing. It's like asking what is so evil about a gun. Nothing. It's the idiot wielding it that has the problem.
See Steve Jobs' Thoughts on Flash: http://www.apple.com/hotnews/thoughts-on-flash/
Look at any major car maker's website, they are ALL in Flash. It depends on what you are doing. If your goal is to provide a rich, sexy website Flash/Silverlight will give you a huge advantage in terms of development time. IF you are providing content/data HTML and JavaScript (jQuery for example) can give you a lot of whizbang without requiring additional software plug-ins (Flash/Silverlight).
I think it's a matter of selecting the lowest cost medium to reach your target audience, be it Flash or Silverlight or JavaScript or plain text.
You can have text-only, JavaScript-only, etc. versions of your site if you have the money, and your target audience come with different prioritizes.
So my question to you would be: does your website has people who frown on Flash as its target audience? If yes, you need to stay away from it. If not, use anything to give your visitors a good experience.