"Spelling checker" or "spell checker"?

I've usually seen it used in combination form: spellchecker. The verb is spellcheck and it conjugates the same way check does. It's also used as spell checker, and I guess you can take your pick.

On the copy of the NOAD installed on the Mac computer I am using to answer, for spelling checker I only read another term for spellchecker. Spell checker is reported as possible variant of spell-checker.

To spell-check (or to spell check) means to check the spelling of a text using a spell-checker; the meaning of spell-checker is reported to be a computer program that checks the spelling of words in files of text, typically by comparison with a stored list of words.

To check the spelling would have a more generic meaning that to spell check, as the latter verb implicates the use of a computer program. It is possible that, as nowadays the computer is used also as text editor, to spell check is used even when the operation has not be done with a computer (e.g., the professor has spell-checked my essay).

Searching for [check] the spelling and [spellcheck] on the Corpus of Contemporary American ([check] matches check, checks, checked) I get the following data (the values are the frequencies per million; the images can also be seen here, and here):

The data evidences that spellcheck is used more frequently in the last 10 years.

Spell check is a little more informal and more modern-ish of a word as a verb, but it is grammatically correct. Usually it refers to Microsoft Office Word or OpenOffice. Checking the spelling is the more uncommon of the two, but still correctly used. (Some people just use spell check instead of checking the spelling because they think it's a waste or precious breath.)

It's not all that difficult , even though our English language is a contradiction of itself . You can call it as a verb , to spell check or spellcheck if one wishes , HOWEVER , based on our building of our language , to spell check is the action verb eg , "Please spell check that document" or "Spellcheck that document."
As a NOUN , there's ONLY one way to spell it ---- it's a spell-check(er) eg , "They performed a spell-check or we need a spell-checker to do its work. " There . Peace.