Minecraft Classic freezes at "switching applet"

The short answer is, you fix this by using a virtual machine. It's an extreme approach, but if your computer is powerful enough it will get you Minecraft Classic, playable in-browser. A virtual machine is basically a computer that runs on your computer.

For a little more detail, you can get software called VirtualBox, which is used to run virtual machines, here. You will need to have some operating system to install on your virtual machine, like an old copy of Windows XP or some free Linux distribution. In the manual for VirtualBox the instructions you'll be interested in start on the 16th page, which show you how to start VirtualBox and set up a virtual machine.

Once you have your virtual machine running, you'll have to install a browser on it if it doesn't already have one, and install Java 6. You can download Java 6 (Java SE Runtime Environment 6u45 to be more specific) here, although you'll have to create an Oracle account to get it...I haven't a clue why, it's just annoying.

If you manage to get to this point, the last step is to start your virtual machine, and then disable mouse integration. In VirtualBox, this can be done by choosing "Disable Mouse Integration" under the "Machine" menu. Disabling this fixes a problem with mousing in Minecraft where your cursor doesn't center automatically, making it nearly unplayable.

If you manage to do all of the above, you should be able to boot up your browser and head over to Minecraft Classic.

Sorry to say this, but unless you play minecraft classic though the time machine in the new launcher, the latest version of Java is not compatible with the browser-based minecraft classic.