Microsoft claims that Internet Explorer versions 4.0 and up are ECMAScript-compliant, along with some extra, proprietary features that are specific to MSIE. So as long as you write ECMAScript-compatible code, it should run just fine under MSIE 4.0+ and Netscape Navigator 6.0+ (and its successors, the Firefox and Mozilla browser families). But you should always test your code with different browsers, platforms, and versions just to be sure.
Apple's Safari has always supported ECMAScript. The current Safari 2.0, which shipped with Mac OS X 10.4 ("Tiger"), supports ECMAScript-262 Edition 3.