My first major surprise after getting engaged occurred at the Shabbos lunch on my first time with my soon-to-be new family. My mother-in-law-to-be offered "regular" fish or gefilte and I, being risk averse, chose the gefilte option. Everyone else enjoyed my wife-to-be's grandmother's authentic Hungarian fish (my mother-in-law's parents live upstairs from them and, at the time, made the fish for the whole family). You can imagine my absolute shock -- hidden only by sheer will -- when everyone started picking bones out of the fish before eating it. I mean, is that not a classic case of the forbidden borer labor, separating "bad" from "good"? What's next, are they going to drive to synagogue? But I kept this to myself and looked into the issue. It turns out that the matter is not so simple and they were simply eating the fish the way their Hungarian ancestors did.
A few months later, Dr. Haym Soloveitchik published a now-classic article (ironically, his most famous article is totally removed from his area of expertise, medieval Franco-German Jewry) in which he mentioned this issue (see here and in notes 2 and 3). I have tremendous respect for Dr. Soloveitchik, and am one of the few people I know who is not terrified of the man, but I don't take his halakhic proclamations as authoritative and he does have some family baggage in this issue, as I'll mention later. I find it surprising that he cites the Hazon Ish's critique of the Mishnah Berurah when it has been shown to be incorrect (see below).
The Mishnah Berurah (319:4) addresses the issue of removing bones from fish in a long Bi'ur Halakhah (sv. mitokh) in which he concludes that there is room to be melamed zekhus -- justify but not advocate -- the practice. His reasons for defending the practice are as follows:
I. What Kind of Bones?
If the bones are soft enough to eat, then there is room to say that they are not "bad" and that one who separates bones from fish is separating "good" from "good", which is permissible.
Furthermore, even with hard, inedible bones, if the bones have some fish or juice on them, one can consider them to also be "good". It would not hurt to, every once in a while, eat the fish meat or suck the juice off of a bone to prove the point.
II. Part of Eating
In the preceding Bi'ur Halakhah (sv. ha-borer), the author quotes the Birkei Yosef who cites a dispute between Mahari Abulafia and Maharit Tzahalon. According to the former, one may separate "bad" from "good" immediately prior to eating the "good", despite the prohibition of borer. The latter prohibits it. The author suggests that the Ramban and the Rosh disagree on this issue, quoting a Rosh that seems to prohibit it and a Ramban that seems to permit it. While we may not follow this ab initio, the lenient opinion seems authoritative enough to justify the practice of those who separate bones from fish while eating.
However, upon reading the Rosh, it seems to me that the Mishnah Berurah/Bi'ur Halakhah is finding a prohibition where there is really the exact opposite -- a permission. See how the Shulhan Arukh (319:16) quotes the Rosh and, even more explicitly, the Tiferes Yisrael (Kalkeles Ha-Shabbos, borer). The explanation that this is not me'ein melakhah is precisely why separating when part of the act of eating is permissible. Certainly the Tiferes Yisrael understands the Rosh as I do, that he is permitting separating when part of the act of eating.
R. Yitzhak Maltzan, in his Shevisas Ha-Shabbos (borer, n. 20), quotes his mentor, R. Hayim Leib of Stavisk, as reading the Ramban differently from the Mishnah Berurah. According to R. Hayim Leib, the Ramban says the exact opposite of what the Mishnah Berurah suggests, prohibiting separating even when immediately prior to eating. However, R. Maltzan found another place in which the Rashba and Ramban seem to permit separating immediately prior to eating.
The Hazon Ish (Orah Hayim 53) rules like the Maharit Tzahalon in disagreeing with the Taz, who permits removing a fly from a cup of wine if one also removes some wine with the fly. Since one is removing both good (wine) and bad (the fly) together, the Taz does not consider this to be separating between the good and the bad. The Maharit Tzahalon and the Hazon Ish dispute this Taz. However, the other posekim overwhelmingly follow the Taz on this issue and it is a major precedent in later halakhic literature.
Therefore, according to this lenient view, one may separate bones from fish either immediately prior to eating the fish or at least when the fish is on the fork and being lifted to one's mouth. While later posekim rule contrary to this view, it is sufficiently well established to justify an existing practice.
III. Connected Items
The Mishnah Berurah/Bi'ur Halakhah proceeds to bring a proof from the Magen Avraham (510:4). The Rema (Orah Hayim 510:2) writes that one can separate a broken nut from its shell if the nut is broken but the food is still connected to the shell. The Magen Avraham quotes the Maharshal who explains that since the shell and the nut are connected, one part cannot be considered "bad" and the other "good". Therefore, separating the two is not considered separating the bad from the good. The Mishnah Berurah makes the natural extrapolation from that case to fish and bones, that are connected. The bones cannot be considered "bad", and therefore one may separate them from the fish meat.
The Hazon Ish (ibid. 54:3) dismisses this suggestion as only applicable to nuts. The case of fish and bones, he contends, is entirely different. Furthermore, R. Hayim Leib of Stavisk (P'nei Aryeh Ha-Hai, cited by Shevisas Ha-Shabbos, ibid.) points out that the entire context of the Magen Avraham and Maharshal is Yom Tov. Perhaps, he suggests, Shabbos is different.
However, R. Shmuel Wosner (Shevet Ha-Levi 1:83) demonstrates that the Hazon Ish's distinction is impossible. The original source of this discussion, R. Yeruham's Toledos Adam Ve-Havah, explicitly applies the ruling to fish bones. Any speculation about distinguishing between nuts and fish bones is, therefore, impossible. Both the Shevet Ha-Levi and the Shevisas Ha-Shabbos point out that R. Yeruham applies his ruling to both Shabbos and Yom Tov. Therefore, R. Hayim Leib's distinction between Shabbos and Yom Tov is also imposible.
The Shevisas Ha-Shabbos also cites the Tel Oros and the Arukh (sv. dash 3), who state explicitly that the prohibition of separating (borer) does not apply to two items that are attached.
Here is how the Shemiras Shabbos Ke-Hilkhasah (3:13) phrases its ruling on this issue:
[T]here are some who are in the habit of removing bones from fish or meat in the normal way on Shabbath and they have rabbinical authorities to support them, but, if this is done, it should only be done in the process of eating and not before.Interestingly, the Shevisas Ha-Shabbos rules leniently more forcefully and concludes that one who wishes to be strict, "a blessing will come upon him." He relates that R. Yosef Dov Soloveitchik, the author of Beis Ha-Levi (great-grandfather of R. Joseph B. Soloveitchik and great-great-grandfather of Dr. Haym Soloveitchik, quoted earlier in this post), when he was given fish with bones, would make sure to eat the bones rather than separate them from the fish! However, the Minhas Shabbos (80) has another tradition about R. Soloveitchik, and writes that he would separate the bones but then make sure to suck some juice off of them.
R. Moshe Feinstein (Iggeros Moshe, Orah Hayim 4:74) permits removing bones from fish when it is impossible to adequately separate them in one's mouth (which I think is the case with my wife's grandmother's fish), and even then only immediately prior to eating.
Nevertheless, after over 11 years of marriage I have still never tried the Hungarian fish and probably never will.