How frequently did Shakespeare reference the rose in any of his plays or sonnets?
I am not exactly sure how to answer this question. The basic answer seems to be that Shakespeare talked about roses quite often, but I'm not sure that that is a very helpful answer.
According to at least one search engine that searches all of Shakespeare's works, he used the word 155 times. I do not know how that compares with other words, but the same search engine says that he used the word "moon" 160 times. (In both cases, the search engine only counts it as one use if the word is used repeatedly in a short time.) That suggests that the image of a rose was one that Shakespeare used quite frequently.