The Shulhan Aruch (Orah Haim 339:4) writes, "En Danim Be’Shabbat" – Bet Din does not convene on Shabbat to try cases. The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) explained that this prohibition was enacted by the Sages out of concern that the judges may write on Shabbat. On the level of Torah law, no prohibition is entailed in convening a Bet Din on Shabbat, but the Sages feared that the judges may write during the trial, in violation of Shabbat, and they therefore forbade convening a Bet Din on Shabbat. It is forbidden even to simply hear the claims of the litigants, as the judges may write notes during the hearing.
The Rama (Rabbi Moshe Isserles of Cracow, 1525-1572), in his glosses to the Shulhan Aruch, adds that it is also forbidden to arrest and incarcerate criminals on Shabbat. This prohibition originates from the time of Babylonian Geonim, and is mentioned by Rav Sherira Gaon, as cited by the Bet Yosef. Incarceration is included under the prohibition against judicial activity on Shabbat, and is therefore forbidden.
The Mishna Berura cites in this context a comment of the Talmud Yerushalmi which finds a source in the Torah for this prohibition. The Torah writes, "You shall not burn fire in any of your residences on the day of Shabbat" (Shemot 35:3), which refers not only to kindling a flame, but also to the fire of capital punishment. By extension, this verse forbids any type of judgment on Shabbat.
In explaining the concept underlying this Halacha, the Sefer Ha’hinuch (anonymous late 13th-century work), in Siman 114 (listen to audio recording for precise citation), writes that on Shabbat, God grants a reprieve to the entire Jewish nation, even sinners and criminals. The Sefer Ha’hinuch compares Shabbat to a banquet hosted by a king, to which he invites the entire citizenry. In honor of the occasion, the king does not start inspecting people’s records to keep out criminals; he reserves this process of judgment until after the banquet. Similarly, on Shabbat, God wants all members of the Jewish nation to enjoy the "banquet," and therefore Bet Din does not convene to judge on Shabbat.
There are, however, two important exceptions to this rule, both of which involve the concern to prevent a woman from becoming an Aguna ("chained"). First, authorities may apprehend and incarcerate a husband who tries to escape from town to avoid giving his wife a Get. Under such circumstances, the Hachamim suspended the prohibition against incarceration, and allowed incarcerating the husband so that he could be forced to give a Get after Shabbat. Likewise, Bet Din may convene on Shabbat in order to receive testimony from a gravely ill witness about a husband’s death. Meaning, if a husband is missing and suspected dead, such as if he went out to war and did not return, and there is only one known witness to his death, if that witness is gravely ill, Bet Din may convene on Shabbat to hear his testimony. In such a situation, delaying the testimony until Sunday could result in the wife being an Aguna, as the witness could die at any moment. Therefore, under these circumstances, the prohibition against convening a Bet Din is waived and the testimony is heard. These two exceptions demonstrate the unique importance that the Hachamim afforded to the concern to avoid situations of Aguna.
Summary: It is forbidden on Shabbat for a Bet Din to convene for any reason, and to incarcerate a criminal. The exceptions to this rule is arresting a husband who is trying to escape to avoid giving his wife a Get, and a situation where only one witness saw a husband’s death, and he is gravely ill and could die at any moment, in which case Bet Din convenes even on Shabbat to hear his testimony.
