Red underlining of a misspelled word has disappeared

Have you tried resetting your iPad. It cures a lot of problems.
Press and hold the On/Off Sleep/Wake button and the Home
button at the same time for at least ten seconds, until the Apple logo appears.

I turned off auto correct and check spelling and saved this post. Then I force quit Safari, came back and I no longer got the red underline either when typing misspelled words.

*This leads me to think that you should also check the settings on your IPad*

Settings>general>keyboard>auto-correction and check spelling. Make sure you have both of those enabled. Go back to the home screen. Then quit the offending app where the behavior started. (double tap home button, recents tray at bottom comes up, tap on app icon until it wiggles, close the app). Go back into the settings, turn the spelling and auto-correction off and then enable them again ... just toggle them off and on.

Then launch the app again and see if it now works. This worked for me when I experienced the same behavior that you described. You may have to restart your iPad and toggle the settings again, but if the hard reset don't work, this fix may do the trick.

I experimented by disabling the spelling settings and sure enough I still got the red underline in Pages. I didn't try Mail but I did try typing this post when the settings were off, purposely made spelling errors, and I did not get the red line either while typing here.

Convoluted, but it worked. Worth a try.

No promises but a system Reset cures many ills:
Hold down the on/off switch and the Home button simultaneously until you see the Apple logo. Ignore the "Slide to power off" text if it appears.

A Reset does not erase network passwords. Only the "Reset networks settings" does that. Also, a Reset does not delete apps, photos, movies, music or any other data.

Finally, you might wish to download the 4.2 User Guide. Pay special attention to page 170.
http://support.apple.com/manuals/#ipad

You might want to do something about your password situation. Sooner or later it's going to bite you.

The type of reset that Philly suggests is worth a try and you will NOT lose anything by doing it. No lost passwords, general settings, nothing like that.

Try toggling the preferences in the keyboard settings for auto correct and spelling off and on. Then reboot the iPad again.

As I stated, it seems convoluted trying my way, but I was actually able to recreate your dilemma and get back out of it the way I described to you and the spell check and auto correct worked again for me. Obviously if it did not work for you, it was of no use to you. You could give Philly's advice a try and then toggle as I suggest.

I also agree that you should take care of your password situation before something really nasty occurs.

Restore from backup via iTunes. Then if still problem restore to factory defaults/new iPod via iTunes.