There's one challenge in coming up with such a test, however. Different books and teachers have different ideas about what constitutes correct English grammar, and there are differences of opinion even about such issues as how to define grammatical terms. That means that no substantive assessment of grammatical knowledge can be completely theory neutral. I've tried to create questions that don't depend too narrowly on a single approach to grammar, but I also won't accept any old assertion just because you can find it in some book or other. I have tried to avoid "gotcha" questions where there are significant disagreements among experts as to what should count as correct. That doesn't mean, however, that you won't be surprised by a few of the answers if you're unfamiliar with modern accounts of English grammar.
The correct answers involving grammatical terminology are those that reflect a mainstream consensus among contemporary linguists, one broadly in keeping with the major recent reference books of English grammar. Unless otherwise indicated, the term "error" is used to mean that the language in question does not conform to the requirements of standard written English, as widely understood by English teachers and editors. The questions will become harder as you move through the quiz. If you have a reasonable knowledge of traditional grammatical concepts, you should be able to get most of these right, although you will need some exposure to linguistics to get a few of the final ones.
Update 1/7/15: Now that several hundred people have taken the complete quiz, I've used the statistics to refine the quiz a bit, clarifying the wording of a few questions and replacing a couple that weren't working the way I'd hoped.