The most common ways to say "my handwriting is bad" would probably be
* **** {} ( often means "dirty," but here's it's "bad" or "messy")
* **** {} ( means "bad" or "unskillful")
So, in that specific case **** is the word you're looking for.
**** refers to the actual letters themselves. You can use for any letters, even ones that are printed, but based on context it will usually be clear if you mean handwriting.
**** is a broadly used term for handwritten letters or the manner of writing. It also appears in compounds like {} (graphology) or {} (handwriting analysis). The fact that it is kind of a "bigger" word than is what probably leads to it being used less in that kind of off-the-cuff remark.
**** is writing or drawing by hand, as well as things written or drawn by hand. It's more about the process, the act of using your hand rather than a machine, than about the resulting letters themselves. Examples include (sign by hand) or (hand-drawn illustration).