You get the best performance and feature set in Chrome. It works fine in Firefox too, but the latest Safari makes problems (I'm almost sure that's Safari's fault, not mine).
In Chrome, copy/paste works, you can drop in files from your desktop, sound output and microphone input works, you can save and load projects, you can switch to fullscreen mode, QuickGuides and translations work, etc.
You can also pass a project URL to directly launch a project, e.g.:
... so it can be used just like the old browser plugin. To satisfy the browser's security model, the project server must either host SqueakJS itself, or support CORS (http://enable-cors.org/server.html).
Performance isn't great yet, but there are students working on it at the U of Potsdam. Some startup time could be shaved off by not providing translation files.