The Oral Torah makes it clear that these laws includes more than just the capture and slaughter of herd animals, such as sheep or goats. The rabbis specifically noted that one who killed abominations and creeping things (rodents, reptiles and insects) “on Shabbat is as though he killed a camel on Shabbat... Rabbi Eliezer holds: It is as the rams: just as there was the taking of life in the case of the rams, so whatever constitutes the taking of life [is a culpable offense]" (Talmud Shabbat 107b).
The sages go on to debate whether this same rule (including all living) applies to the prohibition of trapping as well. “Rabbi Eliezer and Rabbi Joshua disagree only in that one Master holds: If the species is not hunted, one is liable; whilst the other Master holds: He is exempt” (ibid). It is generally accepted that trapping any animal is prohibited on Shabbat
When it comes to the question of trapping and killing animals, an exception is made if one’s life is in danger. “If one catches a snake on Shabbat...so that it should not bite him, he is exempt (from any punishment); if for a remedy, he is liable” (ibid). Additionally, in the case of abominations and creeping things, an exception is made if the creature is causing one bodily discomfort (e.g. a mosquito), in which case one may trap or even kill it.
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