Do you have a dictionary installed and selected?
You can see which dictionary is selected if you right-click in a text area and open the Languages submenu.
You can open the "Add Dictionaries" link to install a dictionary if you do not have one.
Make sure that [X] "Check Spelling" in the right-click context menu has a tick.
You can enable or disable spell checking globally:

• Tools > Options > Advanced : General: Browsing: "Check my spelling as I type"

See also:

• http://kb.mozillazine.org/Spell_checking
• http://kb.mozillazine.org/Dictionaries

Thanks Cor-el,

I had French dictionary but disabled it, but that was not problem. I have spell check on. When I use it, it works as there is a double underline where needed.

Maybe I should uninstall Firefox and all my settings and reload it.

I just had the same problem, but checked the dictionary operating and indeed for some reason, only a French Dictionary was installed! Downloaded English (US) and all seems well. Thanks cor-el! Not to Firefox -- please may we have English (UK) sometime soon?

There are licensing issues that prevent including a suitable dictionary with each locale, so only a few locales come with a dictionary and the en-GB locale isn't among them.