The Shabbat prohibition of Borer forbids removing undesirable items that are mixed together with desirable items. The question arises as to whether it would be forbidden to remove bones from fish as one eats on Shabbat.
The Hid"a (Rav Haim Yosef David Azulai, 1724-1806), in his work Birkeh Yosef, cites a debate among the Aharonim (post-Medieval Halachic authorities) as to whether the Borer prohibition applies to separating Pesolet (undesirable food) from Ochel (desirable food) during the eating process. On the one hand, one might argue that the Borer prohibition applies regardless of when the separation is performed; it is never permissible to separate Pesolet from Ochel on Shabbat. On the other hand, separating during the "hand to mouth" eating process might be considered an integral stage of eating. Since the Torah clearly allows eating on Shabbat, it perhaps allows separating Pesolet from one’s food during eating, as this constitutes an important stage of the eating process. The Hid"a cites the Mahari Abulafia as permitting separating Pesolet from one’s food as he eats, whereas the Rav Yom Tob Sahalon (1559-1638) rules stringently, and applies the Borer prohibition even to separating during the eating process. Of course, according to all opinions it is forbidden to separate Pesolet while preparing food to be served, even just before serving.
The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933), in his Be’ur Halacha, notes that this issue was already debated earlier, by the Rishonim (Medieval Halachic scholars). The Ramban (Rabbi Moshe Nahmanides of Spain, 1194-1270) allowed separating during eating, while the Rosh (Rabbenu Asher Ben Yehiel, Germany-Spain, 1250-1327) ruled stringently.
As for the final Halacha, the Hid"a writes that since a Torah violation of Shabbat is at stake, we must follow the stringent opinion, and refrain from separating Pesolet from food even as we eat. Rabbi Moshe Halevi (Israel, 1961-2001), in his Menuhat Ahaba, likewise rules stringently, but he adds that the lenient ruling of the Mahari Abulafia may be taken into account in situations where there are other considerations warranting a lenient ruling. With regard to removing bones from fish, Rabbenu Hananel (early 11th century) held that since the meat of the fish is attached to the bone, removing the bone does not violate the prohibition of Borer. Furthermore, some authorities held that since fish is normally eaten by first removing the bones before bringing the meat to the mouth, this would not constitute Borer, and is rather the usual manner of eating. Rabbi Moshe Halevi thus rules that in consideration of these arguments, combined with the ruling of the Mahari Abulafia, who permits separating during the eating process, we may allow removing bones from fish while eating on Shabbat. He adds that adults feeding children may separate the bones before giving the food to the children, since the children are incapable of doing so themselves.
Summary: It is permissible to remove bones from fish while eating on Shabbat. One may not, however, separate the bones before serving the fish, except when serving children, who cannot separate the bones themselves.