Female Orthodox Rabbis
a) are totally impossible and an oxymoron
b) couldn't happen because any woman who wants this must have an agenda
c) might in theory be possible but will never happen for practical reasons
d) may happen some day in the future, but not in my lifetime
e) are something we should press hard to create
f) Leave this question out of my results
Some of you may be surprised (outraged?) by my having chosen (c).
The first thing we need to do is to define "rabbi". (Perhaps "orthodox" needs to be defined, too, but I'll risk assuming we're all operating with more or less the same definition of it. And I hope it's not necessary to define "female".) One could argue that there are female orthodox rabbis even now: there are certainly learned orthodox women who write books about Judaism, teach it, travel extensively to lecture about it, are consulted by other Jews for their advice in matters that relate to religion, engage in kiruv, etc. Many male rabbis do almost exactly the same thing (or less), and their own education may not differ very much either. But I don't think that that's what the question means. The question is probably intended to have the answerer consider two specific functions of rabbis that orthodox women really do not perform: serving as a congregational/pulpit rabbi, and acting as a posek. (Note: Some orthodox women certainly do act as poskot of a certain type; I know for a fact that many learned orthodox women will often field questions from their less learned peers about kashrus questions, or niddah questions, or Shabbat questions, etc., the answers to which they know because they are well-read and are familiar with the halachos that they need to know in order to practise Judaism properly. That's not the type of posek I'm talking about, though. I'm talking about the type of posek to whom people turn for rulings on complex issues that have never been ruled on before: a Rav Moshe Feinstein- or Rav Elyashiv- or Rav Hershel Schachter-type posek (covered my bases, didn't I?).)
(Real) Judaism has survived for thousands of years without (any significant number of) female rabbis, and seems to be continuing to do all right despite the modern phenomenon of feminism, so I don't see why (e) should be correct.
The difference between (c) and (d) is purely an issue of fortune-telling. My instinct is that there will never be female orthodox rabbis of the type described, even after I die (***). We're now in the year 5766. If Mashiach is supposed to come by the year 6000, that only leaves about 234 (Can't have a better number than that!) years for orthodoxy to be persuaded to accept and implement the notion that female rabbis are an okay idea. (Once Mashiach comes, all bets are off, as far as I'm concerned.) Is it conceivable that such a thing may happen? I think it is. Is it likely? I think it's not. Can I prove it? No. Can anyone prove otherwise? I doubt it. Orthodoxy is, after all, orthodox; we're a pretty traditional and inflexible lot, and rightly so: we're responsible for preserving something that it is very important to preserve properly and faithfully. I don't think female rabbis are in the cards (although if they were, my vote would be for the Queen of Spades).
As for (a): I have read, in very traditional, "frum" books, that at least according to some great Jewish authorities, (some) women may learn Gemara; and some of them, I fully believe, are smart enough to become talmidot chachamim of the finest calibre. Would such a woman, having achieved such a level of scholarship, not have a duty to paskan? And consider Devorah the prophetess. If there had been shuls of the modern type in her day, might she not have played the role (with a few alterations) of congregational rabbi? I'm not the first one to point out that many or most pulpit rabbis don't do very much, if anything, that a woman isn't allowed to do; indeed, most pulpit-rabbi tasks are ones that some orthodox - even yeshivish - women already do. So is it "totally impossible"? I don't think so. An oxymoron? I don't think our terms are even tightly enough defined for the word to be applicable.
And I think (b) is a silly answer, because it assumes that since certain ambitions tend currently to be associated with a particular (let us assume, unacceptably feminist) agenda, this association will always exist, and will exist in every single case. There have been examples in Jewish history of very scholarly, very devout, and very unsubversive women who have been Talmudic scholars or leaders of Jewry. It would be, I think, a combination of short-sightedness, narrow-mindedness, axe-grinding, blind acceptance of stereotyping and/or pure lack of imagination that would lead someone to assume it impossible for this ever to happen again, just because in our particular era, most (let us assume) orthodox women who want to be/wish they could be rabbis possess that desire because they feel that orthodox Judaism, in its present state, gives them a raw deal, and they want to change things/stir the pot/make a statement.