Gemara

(Mishnah): All Kelim in Yerushalayim on the path people use to descend to the Mikveh are Temei'im.

21b - Question: R. Avahu taught that there was no decree about Safek Kelim in Yerushalayim!

Answer: Since they were found on the path descending from the Mikveh, this proves [that they are Vadai Temei'im].

Nidah 18b (R. Yochanan): In three places, Chachamim considered a majority like a Vadai -- blood of the Makor, a fetal sac, and a piece [of an aborted fetus.]

Question: Chachamim follow the majority absolutely also regarding stores! (If meat is found, it is considered to be Nevelah or Kosher based on the majority of stores in the city.)

Answer: R. Yochanan discusses Tum'ah of a woman.

He already taught about when a Chazakah opposes the majority. (We follow the majority, but we do not burn Terumah due to it.)

Yoma 84b - Suggestion: Shmuel follows the majority regarding Piku'ach Nefesh!

(Mishnah): When a baby is found, we assume that he is like the majority of the city. If half the city are Yisraelim, he is a Yisrael.

(Rav): We follow the majority to support him, but not for lineage (for marriage).

(Shmuel): (We follow the majority) to unearth him.

Rejection: Shmuel comments on the Reisha of the Mishnah. If most are Nochrim, the child is a Nochri;

Shmuel: This does not apply to unearthing him.

(Rav Papa): According to Shmuel, the Mishnah says 'if most are Nochrim, he is a Nochri' to permit feeding to him Neveilos.

Rishonim

Rambam (Hilchos Isurei Bi'ah 15:26): [If an abandoned baby was found, and he did not immerse, and Beis Din did not immerse him,] if most of the city are Nochrim, one may feed to him Nevelos. If most are Yisre'elim, we return an Aveidah to him like to a Yisrael. If it is half-half, it is a Mitzvah to support him, like a Yisrael, and if a house collapsed on him on Shabbos, we unearth him. Regarding damages, he is like every monetary Safek; ha'Motzi mi'Chavero Alav ha'Re'ayah.

Magid Mishneh: Many disagree and say that the Mitzvah to support him is only if most are Yisre'elim. We unearth him even if most are Nochrim. Regarding Piku'ach Nefesh, we do not follow the majority. The Gemara says that if it is half-half, he pays half-damage. This implies that if most are Nochrim, he is considered a Nochri and pays full damage, for he has no Chezkas Mamon, since he himself is unsure whether he is a Nochri or Yisrael. When most are Yisrael, if a Yisrael's ox gored his animal, presumably he need not pay, for we do not follow the majority regarding money. This is why we said specifically returning an Aveidah (according to Tosfos in Yoma and Kesuvos, returning an Aveidah is a bigger Chidush), for the finder has no Chezkas Mamon. If one needed to pay damage when the majority are Yisrael, the Gemara should have taught this, and all the more so one must return his Aveidah!

Milchamos Hash-m (Kidushin 21b): Chachamim did not consider the majority like Vadai when there is a Chazakah against it. The Rif holds that similarly, we are not lenient to do an action whenever it is proper to be stringent, for we join the minority [to the Chazakah]. Rabanan ignore the minority, i.e. it is not important to join it for Taharah, so the majority prevails, just we do not do an action. If a woman went overseas pregnant [and her only son died without children], the majority of outcomes permit the widow to remarry [without Yibum or Chalitzah, i.e. if the fetus is a female or Nefel. Only a viable male forbids her]. Even so, she must be concerned, for her Chazakah was weakened.

Milchamos Hash-m (Chulin 3b): We have a tradition from the Ge'onim that the majority [of those who slaughter are experts] is not like Vadai, for against it is a Chazakah that an animal is forbidden while it is alive. Even though we rely on the majority when it is impossible [not to], this is because it is slaughtered in front of you, and there is no Re'usa (problem).

Aruch l'Ner (Nidah 33b DH Hainu): We burn Terumah due to Safek Tum'ah of spit, only in the upper market, for there Zavim are the majority. We burn due to Safek Tum'ah of Kelim, i.e. on the descent to the Mikveh, for there the majority are Temei'im. Regarding Safek urine, we burn only if there is a majority, e.g. it is next to urine of an animal, so it is probably of a Nochri.

Poskim

Shulchan Aruch (EH 4:34): If [an abandoned baby] did not immerse, if most of the city are Nochrim, one may feed him forbidden food. If most are Yisre'elim, we return an Aveidah to him like to a Yisrael. If it is half-half, one must support him, and we unearth him on Shabbos. Regarding damages and every Safek, ha'Motzi mi'Chavero Alav ha'Re'ayah.

Chelkas Mechokek (32): When most are Nochrim, we consider him a Vadai Nochri. If his ox gored, he pays full damage even if it is Tam, like a Nochri's ox, even though he is Muchzak in his money. Since the Safek is about him, whether he is a Nochri or Yisrael, even regarding money we follow the majority. Likewise, when most are Yisre'elim we return his Aveidah, even though we do not follow the majority regarding money, i.e. to remove it from one who is Muchzak in it b'Heter. One who finds an Aveidah is not Muchzak b'Heter with the owner's consent. Similarly, when most are Yisre'elim, if a Yisrael's ox gores his animal, he is totally exempt. We say 'bring a proof that you are a Yisrael.' We do not follow the majority to take money from one who is Muchzak. The Tur says so, and also the Magid Mishneh according to the Rambam. (The Bach says that in this case, he is like a full Yisrael, and the damager must pay.) Really, there is not a proper majority, since it is Kavu'a. Therefore, one who kills him is not killed, whether most are Nochrim or Yisre'elim, like the Bach himself says.

Rema: Some say that even if most are Nochrim, we unearth him on Shabbos. One is obligated to support him only if the majority are Yisre'elim.

Shulchan Aruch (17:11): If a document was found that says 'Ploni ben Ploni died or was killed', his wife may remarry.

Rema: If we can say that this was written due to rumors that he died, e.g. he drowned in a place from where one cannot see the shore, we do not permit through a document. Perhaps it was written due to the rumors.

Chelkas Mechokek (22): This is not only due to rumors. The same applies if people spoke l'Fi Tumo (unaware of the consequences). People say 'Ploni died' about one who drowned in the river or was killed in war. Terumas ha'Deshen (241) brings a proof from Rav Nachman, who swore that fish ate Ploni, even though this would not permit his wife. Even though he was a great Chacham, Tzadik and Chasid, he swore based on the majority, and what is normal. All the more so Nochrim, who generally lie, rely on the majority and what is normal to consider it Vadai. If so, if a witness testifies based on what he heard from a witness, we must be concerned lest he speaks like people talk.

Aruch ha'Shulchan (163:10): If a woman did Yibum within three months, and gave birth six months later, since most babies are born after nine months, the baby is considered the Vadai heir. "We do not follow the majority in money" applies only when the other is Muchzak in the money. However, the majority is weakened [source the fetus was not recognized after three months]. Therefore, it is like an even Safek.

Shulchan Aruch (YD 28:17): If one slaughtered [a Chayah or bird] and it became Nevelah, or it was found to be Terefah, he is exempt from covering the blood. Similarly, if a Cheresh, lunatic or child slaughtered without supervision, the Chazakah is that their Shechitah is improper, and one is exempt from covering.

Beis Yosef (DH Cheresh): Since most of their deeds are improper, it is like a Vadai Nevelah.

Shach (110:15): If nine stores sell Nevelah, and one sells Kosher meat, and one does not know from which he bought, we do not follow the majority to consider it Vadai Isur. Rather, it is like half-half. It is forbidden only due to Safek.

Binas Adam (Sha'ar Beis ha'Nashim 18:30 DH Od): The Rashba and Mechaber say that after 12 years, a girl is concerned for Kesamim due to Rava's Chazakah [that she brings two hairs immediately]. The Rashba says that if we checked her and found no hairs, we are not concerned lest she had and they fell, since Kesamim are mid'Rabanan. If Rava's Chazakah were like a Safek, even without checking, we may be lenient about a Safek mid'Rabanan! Rather, it is an absolute Chazakah. This is not a proof. Perhaps it is a Chazakah mid'Rabanan. Tosfos often discusses a majority mid'Rabanan. Since they made it like Vadai, she is concerned for Kesamim. Tosfos (Nidah 46b DH R. Yochanan) says [regarding boys] that we lash, for we follow the majority who bring hairs right after 13 years. This shows that it is mid'Oraisa. Others disagree.

Isur v'Heter ha'Aruch (25:12): When there is concern for a Torah Isur, even if mid'Oraisa it is permitted due to the majority, if Chachamim forbade, it is like Vadai Isur.

Minchas Chinuch (34): If Ploni made Reuven a Goses, since most Gosesim die, it should be as if he Vadai killed him. Why isn't he killed in a case when we cannot check [whether or not he will live], e.g. he was killed afterwards or drowned? The Shev Shematsa said that also in capital cases, we follow the majority only if established a Chazakah before the act, e.g. a three-year old girl who became Mekudeshes. The majority establishes her to be an Eshes Ish, or a Ben Sorer u'Moreh [who is established to be an adult through Rava's Chazakah].

Note: Presumably, he says that a three-year old girl becames an Eshes Ish because if she is an Ailonis (who is sterile and does not develop like a normal female), it is Mekach Ta'os (a mistake) which invalidates the Kidushin. We cannot know that a three-year old will be a normal female. The Mishnah says that we kill for Bi'ah with a three-year old Eshes Ish, and we do not say that this is only if her husband stipulated to be Mekadesh her "even if she is an Ailonis.'

Minchas Chinuch (129): A man is Chayav Korban for Bi'ah with a Shifchah Charufah, and she is lashed, only if he completed the Bi'ah. Witnesses do not [normally] see the actual Bi'ah. They merely see them act like adulterers. It is not Vadai that he completed the Bi'ah. If witnesses testify about the Bi'ah, and he says that he did not complete it, he is believed against the majority or Chazakah [that he completed it]. If he does not say so, we rely on this to obligate them.