In this article in Techumin volume 24, Rav Mordechai Ralbag (a חבר בית דין in Yerushalayim) discusses the issue of a single woman who is artificially inseminated (AI) and the status of the child. At the outset, he notes that he is not endorsing the practice, but rather is discussing what happens with the child assuming that a woman has proceeded with said procedure.
Rabbi Ralbag begins by assuming that the most common case of AI is with a married couple who is having fertility issues. He notes that although there is some discussion on the issue, most poskim allow the husband to make his semen available for such a procedure, without worrying about השחתת זרע לבטלה, as there is a clear purpose in this case. However, everyone agrees that so long as the woman is married there is no permission given for another man to donate his semen. There are various reasons given for this prohibition - ranging from an איסור דאורייתא to a range of concerns that will develop down the road to a visceral distaste - but the agreement on the basic law stands.
In terms of the status of the child in such a case, if the donor is not Jewish then the child is fine, as the law is that a non-Jewish man who impregnates a Jewish woman produces a child whose halachic status is clear. If the donor is a Jew, then there is a debate as to whether or not the child has the status of a ממזר.
When it comes to the case of a sperm bank, how are we to deal with the identification of the donor, assuming that the bank keeps such information private? Outside of Israel, we can assume that most donors are not Jewish, and thus certainly for a single girl there will be no problem, since Rav Moshe Feinstein and Rav Yaakov Breisch have already established that we can follow the majority. If the woman is married, then things are more complicated, since what happens if the donor was in fact a Jew? Rabbi Ralbag invokes the rules of כל דפריש מרובא פריש - that since the woman goes to the hospital to be impregnated, then the sperm is considered to be separated from its source, it which case we can assume that it was derived from the majority, which in the case of חוץ לארץ means that it came from a non-Jew.
Based on an argument about the difference between שתוקי and אסופי, Rabbi Ralbag posits three possible views for our case, where the identity of the mother is known but that of the father is not:
1) According to the Maharit, the child is considered a שתוקי since even the mother does not know the identity of the father, and thus the child is a ספק ממזר.
2) According to the Noda BiYehuda, the child is considered to be an אסופי who has a sign on him (namely that his mother is protecting him). Thus, there is no ספק on the mother, and the ספק on the child is lifted by the fact that the child is considered to be פריש and thus is כשר.
3) According to the Yeshuot Yaakov and the Beit Meir, it is possible that there is no problem at all, since, based on a Gemara in Sotah, we learn that זנות is only an issue when there is actual intercourse, which did not take place in the case of AI. As such, it could be that the child is complete כשר. On the other hand, it is possible that since we know that the single mother became pregnant without know who the father is that the child will be deemed a שתוקי.
Rabbi Ralbag brings his analysis to an end by citing the view of Rav Moshe Feinstein (cited earlier on this blog) that ממזרות can only be contracted via an actual act of intercourse, and since there is none in this case, the child will be fine even if the father is himself a ממזר.
In his conclusion, Rabbi Ralbag brings support from both Rav Shalom Messas (Rav of Yerushalayim) and Rav Zalman Nechemiah Goldberg, that his analysis is correct and that if a single woman does receive AI, even though it is not advised at the outset, nevertheless the child will be deemed כשר and acceptable to marry another Jew without question.