Gemara

(Mishnah): Regarding other Shekatzim u'Rmasim [if one wounds them he is exempt].

Inference: If he kills them he is liable.

(R. Yirmeyah): This is like R. Eliezer:

(Beraisa - R. Eliezer): If one kills a louse on Shabbos he is [liable] like one who kills a camel.

(Rav Yosef): It is even like Chachamim. They argue with R. Eliezer only regarding lice, for they do not reproduce [sexually], but all obligate for other Shekatzim u'Rmasim which reproduce! Both of them learn from rams;

R. Eliezer says, just like they were killed [for the Mishkan], one is liable for killing anything;

Chachamim say, just like they reproduce, one is liable for anything that reproduces.

Question: A Pir'osh (perhaps this is a flea) reproduces, yet R. Yehoshua exempts!

(Beraisa - R. Eliezer): If one traps a Pir'osh on Shabbos he is liable;

R. Yehoshua exempts.

Answer (Rav Ashi): Trapping is unlike killing!

R. Eliezer obligates even if the species is not normally hunted, and R. Yehoshua exempts in such a case. However, R. Yehoshua agrees that he is liable for killing them!

121b (R. Yehoshua ben Levi): One may kill any damager on Shabbos.

Question (Rav Yosef - Beraisa): Five animals may be killed on Shabbos - a fly in Mitzrayim, a wasp in Ninveh, a scorpion in Chadayev, a snake in Eretz Yisrael, or a rabid dog anywhere.

This cannot be like R. Yehudah. He obligates for a Melachah she'Einah Tzerichah l'Gufah! Rather, it is R. Shimon, and he permits only these five!

Answer (Rav Yosef): The case is, they are pursuing him. All agree to this.

(A reciter of Beraisos): If one kills snakes and scorpions on Shabbos, Chasidim are displeased with him.

Rava bar Rav Huna: Chachamim are displeased with these Chasidim!

He argues with Rav Huna;

Rav Huna saw a man kill a wasp [on Shabbos]. He asked 'did you kill them all?!' (What you did will not help.)

R. Aba bar Kahana: Once, a snake fell into the Beis Medrash. Niyosi (a Talmid) killed it. Rebbi remarked that it met someone like it.

Question: Did he mean that Niyosi acted properly? [The snake's intent to kill was carried out on itself.]

Or, did he mean that Niyosi acted improperly? [It kills needlessly, and it was killed needlessly.]

Answer - Question (R. Aba and R. Zeira): May one kill snakes and scorpions on Shabbos?

Answer (R. Yanai): I kill wasps. All the more so I kill snakes and scorpions!

Rejection: Perhaps he permits only [to trample them] l'Fi Tumo.

(Rav Yehudah): One may trample spit l'Fi Tumo.

(Rav Sheshes): One may trample a snake l'Fi Tumo.

(Rav Katina): One may trample a scorpion l'Fi Tumo.

[On Shabbos] there was spit in front of the Reish Galusa. He told them to cover it with a Kli.

Aba bar Marsa: There is no need to do so. Rav Yehudah permits to trample spit l'Fi Tumo!

Rishonim

Rif and Rosh (45b and 16:11): One may kill the five damagers in any case. All agree that one may kill other damagers if they are chasing him.

Ran (DH Nachash): Rashi explains that l'Fi Tumo is not that he explicitly stands on it and kills it. Rather, when he walks l'Fi Tumo and there is a snake or scorpion in front of him, he need not avoid it. Rather, he walks over it. If it dies, it dies. He did not intend. Even R. Yehudah forbids Davar she'Eino Miskaven only mid'Rabanan. He does not decree when damage is common. "He tramples it" connotes unlike this. Also, "I kill wasps" connotes that he may intend. Also, this was taught with trampling spit. If it is permitted only without intent, why did Aba bar Marsa tell the Reish Galusa that he need not cover the spit with a Kli, for Rav Yehudah permits to trample spit l'Fi Tumo? They must cover it until someone will (happen to be walk there and) trample it l'Fi Tumo! Rather, one may trample it even with intent. It is permitted only l'Fi Tumo, i.e. he makes it appear that he does not intend. This is like R. Shimon, who exempts Melachah she'Einah Tzerichah l'Gufah. One may not explicitly kill them. Since they do not often damage like the five taught in the Beraisa, we deviate as much as possible from killing them in the normal way.

Beis Yosef (DH veha'Ran): Also the Magid Mishneh wrote like this in the name of the Ramban and Rashba. It seems that the Rambam agrees.

Rambam (Hilchos Shabbos 11:4): On Shabbos one may kill a Chayah or Remes that Vadai bites and kills, e.g. a fly in Mitzrayim, a wasp in Ninveh, a scorpion in Hadeiv, a snake in Eretz Yisrael, or a rabid dog anywhere. One may kill other damagers only if they are chasing him. If they are stationary or fleeing from him, he may not kill them. If he trampled them l'Fi Tumo while walking and killed them, it is permitted.

Magid Mishneh: This is like the Rif. The Ramban and Rashba agree according to letter of the law, but wrote that the Heter of the species listed when they are not chasing him is only like R. Shimon, who exempts Melachah she'Einah Tzerichah l'Gufah. R. Yehudah forbids, for it is not total Piku'ach Nefesh, since they are not chasing him. This is unlike the Rambam, who rules like R. Yehudah. The Rambam permits here. It seems that he holds that it is Piku'ach Nefesh whenever one sees them.

Beis Yosef (OC 316 DH v'Divrei): He explains that the Rambam explains 'this is when they are chasing him. All agree' means that R. Yehoshua ben Levi permits other damagers when they are chasing him, and this is according to all Tana'im. The Beraisa permits five damagers even when they are not chasing him. R. Yehudah agrees, for they often damage, so it is Piku'ach Nefesh whenever one sees them. It seems that the Rambam similarly permits any Chayah or Remes that Vadai bites and kills. The Gemara said that we permit other damagers only if they are chasing him, i.e. when it is not Vadai that they kill, rather, it is only a Safek. Therefore, when they are chasing him it is Safek Piku'ach Nefesh. If not, R. Yehudah would not permit, for he obligates for Melachah she'Einah Tzerichah l'Gufah. However, the one who holds like R. Shimon could permit even without Safek Piku'ach Nefesh, merely to avoid pain. The Mordechai (402) says 'other damagers, for which there is no danger, one may kill them if they are chasing him. [R. Tam says so,] for he holds like R. Shimon, who exempts Melachah she'Einah Tzerichah l'Gufah. Due to pain, it is permitted l'Chatchilah, just like bursting an abscess to let out pus or extinguishing a coal to avoid injury to people.'

Tosfos (121b DH b'Ratzin): Rashi explains that R. Yehoshua ben Levi permits when they are chasing him, so it is Piku'ach Nefesh. The Beraisa is like R. Shimon, when they are not chasing him. This is difficult. If so, the Gemara should have challenged Rabah bar Rav Huna from the Beraisa, for he permits killing snakes and scorpions on Shabbos. We must say that this is when they are not chasing him, for if they are chasing him, why are Chasidim displeased? This is Piku'ach Nefesh! Also, why was the Gemara unsure whether it was proper for Niyosi to kill the snake? We can learn from here that when they are not chasing him, R. Shimon permits only the five! Also, 'I will explain it' connotes that he comes to explain the Beraisa. Rather, the Ri explains that the Beraisa discusses when they are chasing him, and all agree to it. Even R. Yehudah permits killing the five when they are chasing him, and forbids the others even when they are chasing him. R. Yehoshua ben Levi holds like R. Shimon, who permits the rest, even when they are not chasing him. Rav Huna, who forbids, argues with R. Yehoshua ben Levi. The Ri and R. Chananel say that one should not be lenient like R. Yehoshua ben Levi. Rather, we hold like the Amora'im who permit only l'Fi Tumo.

Poskim

Shulchan Aruch (OC 316:10): On Shabbos one may kill a Chayah or Remes that Vadai bites and kills, even if it is not chasing him.

Beis Yosef (DH Chamishah): Rashi explains that in the Hava Amina, R. Yehoshua ben Levi holds like R. Shimon, who forbids Melachah she'Einah Tzerichah l'Gufah only mid'Rabanan. He did not decree to forbid killing damagers, even if they are not chasing him, for Stam, they kill. According to R. Yehudah, one is liable for Melachah she'Einah Tzerichah l'Gufah, so it is permitted only for Piku'ach Nefesh, i.e. if they are chasing him.

Beis Yosef (DH v'Da'as): The Rif, Rosh, Rambam and Tur hold like Rashi.

Mishnah Berurah (44): This includes things like rabid dogs.

Kaf ha'Chayim (91): The Shulchan Aruch did not list the five damagers, for he holds like the Rambam, that the same applies to similar damagers. The Levush and Gra agree.

Kaf ha'Chayim (92): In the Zohar, R. Chizkiyah wanted to kill a snake, and R. Eliezer told him not to. Perhaps this is only in their habitat, i.e. the Midbar. Alternatively, this was Midas Chasidus. Alternatively, R. Eliezer knew that that snake was going to fulfill its mission, so we may not learn from here to other cases.

Mishnah Berurah (45): This is even if it is fleeing from him.

Shulchan Aruch (ibid.): One may kill other damagers, e.g. a snake or scorpion in a place where they do not kill, only if they are chasing him. If not, it is forbidden.

Gra (DH Kol): Even though the Gemara mentioned only five, the Heter is not limited to them. Everything is according to the time and place, like the Rambam says.

Mishnah Berurah (46): The Rambam obligates for Melachah she'Einah Tzerichah l'Gufah, so he permits only for Safek Piku'ach Nefesh. We forbid Melachah she'Einah Tzerichah l'Gufah only mid'Rabanan, so we permit even to avoid pain. This refers to things that bite painfully. Regarding fleas and small Remasim, which do not hurt so much even when they bite, one may not kill them even when they bite him. Rather, he expels them from himself.

Shulchan Aruch (ibid.): One may trample them l'Fi Tumo, even if he intends, but makes it look like he does not intend.

Magen Avraham (23): Even the Rambam permits this. What is the reason? It seems that he permits only snakes and scorpions, which usually kill. Therefore, even where they do not kill one may trample them due to concern for mortal danger. Surely the Rambam forbids something that never kills.

Kaf ha'Chayim (99): Even if they sometimes kill, Safek Piku'ach Nefesh permits. However, the Shulchan Aruch permits even without danger.

Magen Avraham (23): Some people kill spiders. This is totally forbidden. Spiders do not damage. Even though there is concern lest they fall in food, this is uncommon, and one can cover the food. Also, it is dangerous in food only once in 1000 times. Therefore, one should protest against these people.

Mishnah Berurah (48): All the more so, one may not kill other Remasim and worms found in Peros.

Kaf ha'Chayim (100): One may trample spiders l'Fi Tumo.

Mishnah Berurah (47): Likewise, one may cover them with a Kli.

Mishnah Berurah (48): He may not overtly kill them, lest onlookers say that he intentionally killed it, and they will not know to distinguish this from other cases. One may not kill ants and other Shekatzim u'Rmasim even through walking.

Kaf ha'Chayim (98): One must be careful to avoid killing other Remasim through trampling.

Kaf ha'Chayim (99): A Ba'al Nefesh should not rely on the Heter to trample, when it is not necessary.