I do not believe that a signature must be legible. A signature is just a mark that is used to indicate that you agree to whatever the document is about, and as a result, each person should have a unique signature, even if they share their name with someone else. In fact, when I lived in France, it was pointed out to me several times that my signature was too legible, which they believed made it easier to forge. French signatures tend to be illegible, artistic things, usually composed of the persons initials overlapping each other.
That said, my signature is really not legible, although it does include both my first and last name in cursive. If you knew my name, you could read the signature. If not, then you wouldn’t have a clue what it said.
I use my signature on documents that have my name spelled out elsewhere, like checks, my driver’s license, credit cards, business letters, etc.
If I am “signing” something that requires that my name be legible (like a student registration card, to prove that I advised the student), I usually just write my name in my normal handwriting, which is perfectly legible when it needs to be.
On the rare occasion where I have to actually use my signature, but my name is not already on the document somewhere, I will scrawl the illegible signature, then write my name out next to or under the signature.
However, I do training courses, and I frequently need a list of names of the people who attended a given course, so that they can get credit for attending, or so that I can forward information to them. I usually do not have a printed roster ahead of time, so the sign-in sheet is often the only list I have of who actually attended the course. When they only include an illegible signature, I can’t give them any kind of credit for being there.
I also teach classes at a university, and I do come across the occasional student who feels that I should be able to figure out whose homework I am grading based on a totally illegible scrawl. After the first couple of warnings that I can’t give credit if I can’t read the name, most of them figure out that they should write their name legibly, rather than with their official signature.