Wearing dirty, worn, or frayed anything is frowned upon... Though I don't mind giving folks like you the patch business Janet.
Personally I used to think it was some kind of "cool dude" thing when folks wore their old torn up stuff to class. I guess it gave them an "aura of experience" or something like that.
Probably the belt is the most acceptable form of 'old and frayed' in the dojo. FWIW I remember K. Maruyama Sensei saying that this can easily become a form of vanity.