Filling out online applications should be treated every bit as seriously as your own resume. You wouldn't gloss over grammar and spelling with your resume, so don't do it with online applications. And don't think online applications are read over lightly. Considering you're answering the same questions as others, it's that much more important to make sure your application stands out with proper grammar and correct spelling. Since there's no face-time involved, online applications stand primarily on the power of words.