If one's fingernail got partially unattached, one should trim the whole nail, since otherwise it would constitute an interposition (חציצה) when doing Netilath Yadayim (the ritual washing of hands). If this occurs on Shabbath, the situation is as follows:
If the majority of the nail became unattached, one is permitted to remove the nail, by hand, on Shabbath. One should not use any instrument, such as scissors or clippers, however, because it is a Rabbinical prohibition to do so.
If only a small portion of the nail became unattached on Shabbath, one should not remove it, because it is prohibited by the Rabbis. However, if one did remove it by hand, there would be no obligation to bring a sin offering. If one used an instrument, one would have had to bring a sin offering at the time of the Beth HaMiqdash (Temple), because it is considered to be a subset of the Torah prohibition of shearing (Gozez).
(See Shulhan 'Arukh, 328:31. Mishnah Berurah, ibid., 96-98. Ben Ish Hai, Ben Ish Hai, 1st year, Parashath Aharei Moth/Qedoshim, Oth 24)
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