<p>I'm currently using and enjoying using the Flex MVC framework <a href="http://www.puremvc.org" rel="noreferrer">PureMVC</a>.  I have heard some good things about Cairngorm, which is supported by Adobe and has first-to-market momentum.  And there is a new player called Mate, which has a good deal of buzz.</p>

<p>Has anyone tried two or three of these frameworks and formed an opinion?</p>

<p>Thanks!</p>

Mate
is my pick. The first and foremost reason is that it is completely unobtrusive. My application code has no dependencies on the framework, it is highly decoupled, reusable and testable.
One of the nicest features of Mate is the declarative configuration, essentially you wire up your application in using tags in what is called an event map -- basically a list of events that your application generates, and what actions to take when they occur. The event map gives a good overview of what your application does. Mate uses Flex' own event mechanism, it does not invent its own like most other frameworks. You can dispatch an event from anywhere in the view hierarchy and have it bubble up to the framework automatically, instead of having to have a direct line, like Cairngorms CairngormEventDispatcher or PureMVC's notification system.
Mate also uses a form of dependency injection (leveraging bindings) that makes it possible to connect your models to your views without either one knowing about the other. This is probably the most powerful feature of the framework.
In my view none of the other Flex application frameworks come anywhere near Mate. However, these are the contenders and why I consider them to be less useful:
PureMVC actively denies you many of the benefits of Flex (for example bindings and event bubbling) in order for the framework to be portable -- a doubious goal in my view. It is also over-engineered, and as invasive as they come. Every single part of your application depends on the framework. However, PureMVC isn't terrible, just not a very good fit for Flex. An alternative is
FlexMVCS
, an effort to make PureMVC more suitable for Flex (unfortunately there's no documentation yet, just source).
Cairngorm is a bundle of anti-patterns that lead to applications that are tightly coupled to global variables. Nuff said (but if you're interested,
here are some more of my  thoughts
, and
here too
).
Swiz
is a framework inspired by the Spring framework for Java and Cairngorm (trying to make up for the worst parts of the latter). It provides a dependency injection container and uses metadata to enable auto-wiring of dependencies. It is interesting, but a little bizzare in that goes to such lengths to avoid the global variables of Cairngorm by using dependency injection but then uses a global variable for central event dispatching.
Those are the ones I've tried or researched. There are a few others that I've heard about, but none that I think are widely used. Mate and Swiz were both presented at the recent 360|Flex conference, and there are videos available (
the Mate folks have instructions on how to watch them
)