It rather looks like a problem with your Firefox profile - the Inspect
feature works well on my machine (even with Firefox Nightly builds).
Try to create a new Firefox profile and install just Firebug in it (to
avoid collision with another extension)
http://getfirebug.com/wiki/index.php/FAQ#Installing_in_a_clean_profile
You can also try to reset Firebug options. Firebug icon menu ->
Options -> Reset All Firebug Options.
Honza