Most of the animated effects on this site are realized with Javascript and not Flash. There are several reasons for that:

• Allthough Flash makes your site look more dynamic, it actually makes it a lot slower. The page load times for the same effect made through Javascript are much faster. Probably you noticed that most Flash applications in a website provide a progressbar that shows the state of loading to the impatient visitor. Javascript is faster because it runs from the computer of the user (client side) and not on the server side.
•  Javascript comes with your browser and is therefore more widely supported then Flash Player.
• With flash the content can’t be indexed, so it can’t be crawled and found by search engines.
• Most interesting Flash applications are not Open Source. Not even for non-commercial use and mostly the license is valid only for the site it appears on. That means that applying them in your website has an additional cost.

There are many other reasons that make Javascript favourable against Flash (another article).

Nevertheless if applied in the header banner, Flash can give your site that nice dynamic swing. The header is an elements that only loads once and appears on each next page very fast, without having to load again. The header space is also very valuable for the search engines, so this is only an option if you care less about your ranking on Google. We use Flash, but in an intelligent way. The meanest animations on the web are made with Flash (an example).