Both of my Java pages (Stratego, Numerology) are XHTML compliant and run in Firefox (Windows and Linux), IE, and Safari in Mac. They use the proper XHTML "object" tag rather than the "applet" tag. Based on my web searches, it is rare for a page and applet to work under those conditions. I stumbled onto the proper XHTML by trying to read the directions and experimenting with what others had done. It's also possible that the situation has improved with later versions of all the browsers. (In other words, full compatibility and compliance wasn't possible until recently.)
Everything works great with the Sun / Oracle JRE. Note that you need a JRE installed on your machine for Java to run. As I recall, the IcedTea interpreter in Linux was more erratic. Sometimes I'd have to refresh a page several times or otherwise beat on my system a few times to get the applet to work.
To be specific, my tests were on Windows XP SP3 on Firefox (3.6.10), IE (8.0.6001...), Chrome (6.0.472.63), and Safari (5.0.2); Mac OS X (10.6.4) in Safari 5.0.2 (Mac Mini Aluminum (4,1)). As I recall, when the Sun / Oracle JRE was installed on Linux, everything worked.
Internet Explorer. XHTML 1.0.