Testing desktop web applications means either having a fixed browser, typically the corporate standard, you target or testing across multiple browser versions. While developing mobile web sites drastically reduces the fragmentation inherent to native applications, there are still various browsers to test against. So its important you have the tools available to test.
Believe it or not Opera is the dominating mobile web browser with over 125 million mobile installations. It is also the browser included for many other consumer devices such as the Nintendo Wii and newer smart Televisions.
I have been a fan of Opera for years on the desktop, but neglect to use it consistently like most others. Opera tends to be on the leading edge by integrating support for the latest web standards and proposals. The browser is quick and snappy.
While the desktop Opera browser has a mobile emulation feature, I prefer the Opera Mobile Emulator.
The iPhone definitely pushed the consumer market into the SmartPhone age. Because it is an Apple platform the phone uses the Safari browser. To get the Safari emulator you must download the entire 3.7 GB iOS SDK and of course have an Apple desktop to develop for the platform. So unless you are a MAC person and I am not, you need to fork over around $1000 to develop for the iPhone and test your mobile web sites in mobile Safari.
You can always test your sites in Safari for the desktop, but I really like having a mobile emulator to see what it will feel like on a device. Fortunately Safari is a WebKit browser so you can get a reasonable idea what your site will do by using Chrome or the desktop version. I recommend resizing the window for a more realistic experience.
Here is a great desktop browser that just does not have a good mobile story. There are mobile versions of FireFox available for Androoid and Maemo (a Nokia platform). I just do not see much energy behind a mobile Firefox so I am not real sure how important testing against Firefox mobile is.
Finally the Internet Explorer story. On WinMo 6.5 and prior a version running the IE 6 rendering engine is used. That did not improve much for Windows Phone 7 because it currently uses the IE 7 engine. I call it shipping a phone with a crippled browser. Later this year Microsoft announced it would be updating the browser to support HTML5, which leads me to believe they will rectify this problem with a mobile version of IE 9.
You can test your sites against Windows Mobile IE 6 using the Windows Mobile Device Emulator. I am not sure how much time I would spend testing things against this platform as it and many of the mobile platforms from the pre-smartphone generation are being phased out and are not good data devices. In other words consumers using Windows Mobile and older phones just don't spend too much time browser because it is not an easy experience so the demand is very low.
A great list of mobile emulation environments is maintained at MobilexWeb.com. I think developing great mobile web experiences requires having access to the mobile browsers and emulators to test. I have said it before, I believe architecting your web applications to the web standards is the first strategy for success. Next use progressive enhancement to give your users the best chance of being successful.
I am firm believer you should create a mobile web site for every desktop web site you create these days. The visual context is a completely different experience and form factor from the desktop. This means you must test your applications in an appropriate context. Having a proper library is the best strategy. I have given you a quick overview of the important browsers, where to find them and even a great resource to keep up to date on these and more mobile environments.