I would have thought that handwriting is fine if it's a paper form. Not sure how you'd type up an pre-printed application form without having a manual typewriter, and even then it's not easy to get alignments etc correct.
Make a load of copies of the form before you start filling it out so that you can get any mistakes out of the way, and get someone else to read over it/spell check it before writing out the final version
That being said, if you have a soft copy of the form in editable format, I'd usually type the information into those, being careful not to screw up the page layout.