Question: 6 Jan 2012 v.puzzled, Ubuntu 10.04 Lucid Lynx, Firefox 3.6.24, sun-java6-plugin 6.26-2lucid1:

Firefox does not seem to even know that the java plugin is there (it is installed, and re-installed), i.e., it does not show up in Firefox's tools -> addons -> plugins, nor does it run. (It did, until some recent update.-)

Answer: 
Try installing the open source java plugin `sudo apt-get install icedtea6-plugin`

It works for me.
(Solved for me)
Update of my recent post:

I had the same problem as reported by lloyd allison after the recent update on the Ubuntu 10.04 64-bit system. The problem was compounded but the presence of the NoScript (2.2.6) plugin (which I like ;).

The solution was:

a) disable NoScript and stop firefox
b) install icedtea6-plugin:

sudo apt-get install icedtea6-plugin

c) start firefox, enable again the NoScript plugin, and restart firefox as recommended.

After this, Java plugin starte working again (tested with JMol Java plugin).

Thanks to all who helped.