The prohibition of Borer forbids separating Pesolet – an undesirable substance – from Ochel – desirable food – on Shabbat. This Halacha gives rise to the question of whether it is permissible to eat fish on Shabbat. When eating fish, one generally has to remove the small bones from the fish on his plate. Placing the fish in one’s mouth before removing the bones can, obviously, pose a serious risk to one’s life. Seemingly, it should be forbidden to eat fish on Shabbat, as Halacha forbids remove inedible substances – such as bones – from food on Shabbat.
The Halachic authorities address this question and point to several reasons for allowing eating fish on Shabbat. First, Rabbenu Hananel, cited by the Aruch, was of the opinion that the prohibition of Borer applies only when two substances are mixed together, but not when they are attached to one another. Thus, for example, it is permissible to remove a banana peel in order to eat a banana on Shabbat, and this does not constitute Borer, because the peel and the fruit are not mixed together, but rather attached to one another. Similarly, since the flesh of the fish is attached to the bone, it would be permissible, according to this view, to remove the bone, and this would not violate the prohibition of Borer.
Furthermore, the Be’ur Halacha (Rav Yisrael Meir Kagan of Radin, 1839-1933) cites the view of the Mahara Abulafia that it is permissible to remove Pesolet from Ochel on Shabbat while eating. This view would certainly allow removing bones from fish on one’s plate as he eats. Although Halacha generally does not follow this view, we may take it into consideration along with other factors to allow removing bones from fish. There is also a view that the laws of Borer apply only to food grown from the ground, and not to other foods, such as meat and fish. This view, too, is not generally accepted as Halacha, but it gives rise to yet another Safek (Halachic uncertainty) that may be taken into account.
In light of these considerations, Hacham Ovadia Yosef, in Hazon Ovadia (vol. 4, p. 196; listen to audio recording for precise citation), rules that it is permissible to remove bones from fish on Shabbat. This is also the ruling of the work Shemirat Shabbat Ke’hilchatah, and of many other Halachic authorities. There are those who claim that the common practice among Ashkenazim to eat on Shabbat “gefilte fish,” which does not contain bones, stems from the concern for the prohibition of Borer. According to the accepted Halacha, however, this is not necessary, as it is permissible to remove bones from fish on Shabbat.
It should be noted, however, that this ruling applies only to bones that are still attached to the flesh of the fish. When it comes to bones that have already become detached, the prohibition of Borer applies and one may not move them away from the fish on Shabbat.
Summary: It is permissible to remove bones from fish on one’s plate on Shabbat, and this does not violate the prohibition of Borer.