Gemara

(Mishnah): Mid'Oraisa, a Shetuki (one who recognizes his mother but not his father) is Kosher. It says "Lo Yavo Mamzer (bi'Khal Hash-m)." A definite Mamzer is forbidden, a Safek Mamzer is permitted;

A Mamzer may not marry a Vadai member of Kehal Hash-m (someone of proper lineage), but he may marry a Safek.

(Rava): Mid'Oraisa, a Shetuki is Kosher. This is because most possible fathers are permitted to a single woman. If he came to her (to have Bi'ah) we assume that he came from the majority. Even if she went to him, so the Isur (of men forbidden to her) is considered Kavu'a (fixed), and we view it as an even Safek, the Torah says "a Mamzer will not enter Kehal Hash-m." A Vadai Mamzer may not enter the Kahal, but a Safek Mamzer may. A Mamzer may not marry a Vadai member of the Kahal, but he may marry a Safek member.

The Mishnah forbids a Shetuki to marry Kosher lineages. This is a stringency of lineage (we are stringent only about people of good lineage).

74a (Mishnah): Aba Sha'ul would call a Shetuki (one who does not know who his father is) 'Beduki' (checked).

This means that his mother is believed to say that his father is Kosher.

R. Gamliel taught so in another Mishnah. One might have thought that she is believed only when most men in the city are permitted to her. Aba Sha'ul teaches that she is permitted even when most are forbidden to her.

(Mishnah): All who may not marry into the Kahal, are permitted to each other;

R. Yehudah forbids.

R. Elazar says, Vadai (definite) Mamzerim may marry Vadai Mamzerim. Safek Mamzerim may not marry Vadai or Safek Mamzerim.

75a (Rav Yehudah citing Rav): The Halachah follows R. Elazar.

Rishonim

Rif and Rosh (31b and 4:13): The Halachah follows R. Elazar.

Rambam (Hilchos Isurei Bi'ah 15:12): If a single girl became pregnant and said that it is from a Mamzer, even if the Mamzer agreed, the child is a Safek Mamzer. We say that she was Mezanah also with others.

Ran (30b DH Masnisin): The Rambam holds that she is believed to be Machshir the child, but not to Vadai disqualify him and permit him to a Mamzeres. This is because mid'Oraisa a Shtuki is Kosher. The Torah believed a father about his firstborn, even to disqualify him, but it does not believe the mother. A midwife is believed to identify the Mamzer when women gave birth together and we know that there was a Mamzer. The Torah said that a mother is believed to be Machshir her Shtuki. It did not say 'to be Machshir or disqualify.'

Rebuttal (Shiltei ha'Giborim Kesuvos 5a): Just like she is believed to be Machshir the child, she is believed to say that he is a Mamzer Vadai and permitted to a Mamzeres, since it is known that she became pregnant through Zenus.

Support (Beis Shmuel EH 4:39): We cannot say that she is believed only to be Machshir because the Torah permits a Safek Mamzer, for the Torah permits a Safek Mamzer also to a Mamzeres! Do not say that we are concerned lest he marry a Shifchah, an Isur mid'Oraisa, for the Rambam holds that the Isur of a Shifchah is only mid'Rabanan!

Magid Mishneh: Mid'Oraisa, a Safek Mamzer may marry Kosher Yisraelim but not Mamzerim. Therefore the mother is believed to be Machshir her child, but not to disqualify.

Rebuttal (Beis Shmuel 39): The Torah permits a Safek Mamzer also to Mamzerim! The Prishah (37) says that Chachamim did not believe her to permit him to a Mamzeres, lest this increase Mamzerim. This requires investigation.

Defense (Beis Meir 4:26 DH Ayen): Also the Nimukei Yosef (Yevamos 23b DH Gemara) says that she is believed only to be Machshir because her Chazakah supports this. However, the Magid Mishneh is correct. The Rambam (Isurei Bi'ah 15:19) says that even a married woman is believed if her husband says that it is not his son and she says that it is from a Nochri. Perhaps R. Tam, who says that she is not Chayav Misah for Bi'as Nochri, would say that she keeps her Chezkas Kashrus, but all R. Tam's colleagues disagreed.

Ritva (73a DH Omar): According to Rava, why is a verse needed? A Sefek-Sefeka permits! We need a verse when we know that she went to him and there is only a single Safek, or when half the city is Mamzerim, or for an Asufi.

Beis Meir (DH u'Leyashev): The Ran says that mid'Rabanan, R. Elazar is stringent to forbid Safek Mamzerim to Vadai Mamzerim. The Rambam (15:21,22) does not call this a stringency. He holds like the Ritva, that R. Elazar holds that the Torah permits a Safek Mamzer only to Kesherim.

Poskim

Shulchan Aruch (EH 4:26): If a single girl gave birth and she is not here to check her, or she is a lunatic or mute, and even if she says that the child is from Ploni the Mamzer, the child is a Safek, even if Ploni admits that he had Bi'ah with her.

Chelkas Mechokek (25): We do not believe her to make the child a Vadai Mamzer against the Chazakah that a woman checks before being Mezanah that the Bo'el is Kosher. This is only to be Machshir, but not to disqualify. I say that even after this Chazakah, it is like an even Safek (when the majority of men are Pesulim). Also, perhaps she was raped!

Beis Shmuel (40): It seems that even if there are no grounds to say that she had Bi'ah with others, we are concerned that she did (so the child is not a Vadai Mamzer). The Rambam says similarly in Hilchos Yibum (3:4). However, in Hilchos Terumos (8:14) the Rambam is not concerned lest she had Bi'ah with another! The Bach (156 DH v'Chosav Od) says that the Rambam is concerned only when an Isur mid'Oraisa could result. However, here there is no Isur mid'Oraisa! It seems that the Halachah is that we are concerned only when an Isur mid'Oraisa can result. The Rosh and Rashba are not concerned even regarding an Isur mid'Oraisa. However, according to some Poskim, there is concern for an Isur mid'Oraisa, lest he marry a Shifchah.

Gra (56): The mother is believed in the first seven days only about who is the Bechor.