I have noticed that both Firefox/Outlook don't have auto spellcheck for the subject text box of emails. For Firefox, I am referring to Yahoo web mail.
The subject is supposed to contain texts; then why isn't any spellcheck enabled for these fields?
In Microsoft Outlook 2007 the spell checker does check the content of the subject line. It does not do this as you type, so you don't get the squiggly lines under the misspelled words. You can configure Outlook to check spelling before you send the message.
I expect this will work for other versions of Outlook. It did work in Outlook 2003.
Yahoo mail does not have a spell checker. Firefox will check the spelling of text typed into fields with more than one line of text (see this page). The subject line in a Yahoo mail form appears to be a single line of text, so Firefox doesn't check it. The body of the message is multiple lines, so Firefox does check it.
Update (February 14, 2016)
Firefox can spell check single line text fields. It's just not the default. To change the setting you will have to edit the configuration as follows.
The possible values for layout.spellcheckdefault are described here.
I recently found that sometimes Firefox does not do spell check automatically. I have to right click in a box and select check spelling.
Check this: Disable spell check feature in Firefox