Single women, approaching or past 40, hear their biological clock alarm ringing with tremendous volume. One that I know has decided that her love life is not going to come around soon enough for her to bear children, so after deep thought, she had herself artificially inseminated through at a clinic that uses sperm from anonymous donors. Assuming that DNA tests were done to ensure that the donor was not related to her, was her choice permissible under Jewish law? Even if permissible, would her choice be considered moral?
Rabbi Aaron Rakeffet-Rothkoff has an excellent lecture on the subject, as does the bein din l'din blog.
From a very narrowly halachic perspective, if authorities such as Rabbi Shlomo Zalman Auerbach and Rabbi Moshe Feinstein allowed a married woman whose husband was infertile to receive donor sperm, it should be far simpler, at first glance, to allow the same to an umarried woman. (With the caveat [medically regardless, halachically if the donor is Jewish] that a child of an anonymous donor should use DNA testing to make sure someone they're dating is not actually a half-sibling!)
Broader questions do arise, though; as a culture we value marriage. Rabbi Shlomo Aviner similarly recommends adoption instead -- something to the effect of "why not help out an existing troubled child?"
As Rabbi Rakeffet describes it, in more yeshivish circles this isn't discussed much, although some in this camp dismiss it out-of-hand as yet another feminist push of left-wing Orthodoxy.
The more right-wing Religious Zionist school (such as Rabbi Aviner) decry it, mostly for social reasons from what I understand. A group of rabbis proclaimed it was prohibited a few years ago, with a notable exception of Rabbi Yuval Cherlow, who indicated the matter should be decided on a case-by-case basis with a posek.
Rabbi Rakeffet concludes that "he has a sickening feeling" that the practice will become more common -- and accepted -- in at least some Orthodox circles as the years go by.