The phrase "illegible handwriting" means that someone's writing is hard or impossible to read because it is messy, unclear, or hard to understand. Full definition
The 2004/05 letters are known as the «black spider» letters because of his virtually illegible handwriting. (politics.co.uk)
That misinformation could come from police officers making innocent mistakes like illegible handwriting, or purposeful evasions like giving the wrong name. (politico.com)
The touch - sensitive screen recognizes nearly illegible handwriting. (scientificamerican.com)