Everyone agrees that if a person has a child from a Nidah or a Sotah that the child is not a Mamzer. (1)
When we have a choice to learn out from one of two Hekeshim we learn out from the Hekesh that teaches a stringency.
If a Shomeres Yavam accepts Kidushin from someone other than the Yavam the Kidushin is not valid. (2)
R. Simai says that according to R. Akiva a child born from a Chayavei Lav is a Mamzer with the exception of a child born from the union of a widow to a Kohen Gadol (3)
R. Yeshavev holds that according to R. Akiva even a child born from the union of a widow to a Kohen Gado is a Mamzer.
It is a Machlokes if a child born from an Isur Aseh is a Mamzer according to R. Yeshavev
Everyone agrees that according to R. Akiva a child born from a union of a Be’ulah to a Kohen Gadol is not a Mamzer. (4)
A Kidushin with a non- Jew or non-Jewish maidservant is not valid and a child born from that union is a Mamzer.
A child born from the union of a Jewish woman with a non-Jew is regarded as a Jew; however he is regarded as Pasul.
R. Shimon holds that we Darshen the reason for a Pasuk, while the Rabanan disagree.
A BIT MORE
1.Shimon ha’Teimani who holds that a child born from an Chyavei Krisus is a Mamzer admits that a child born from a Nidah is not a Mamzer and R. Akiva who holds that children who are born from an Isur Lav is a Mamzer agrees that a child born from a Sotah is an exception
2. Even according to the Tana’im who hold that Kidushin with a Chayavei Lav is valid a Shomeres Yavam is an exception.
3. The Torah states that a child born from the union of a widow to a Kohen Gadol is a Chalal and the implication is that he is a Chalal but not a Mamzer.
4. Because it is an Aseh that doesn’t apply to anybody other than a Kohen Gadol.
A HATED WIFE
The Gemara says that according to the Rabanan the case of one hated wife and one loves wife is not literal because Hashem does not love one and hate the other. Tosofos asks why don’t we say that one if them is a Reshah and therefore Hashem hates. Tosfos answers that the there is no reason why the son should lost his inheritance because his mother is a Reshah. However, if the mother is a Chayavei Lava there is reason to think that he son will lose his inheritance because I might have thought that that the son is not a 100% legitimate son.
If someone is Mekadesh a Kussis or maidservant it is not effective because they are not Bnei Kidushin and so too a Kusi or a servant who is Mekadesh a Bas Yisra’el it is not effective. A Yisra’el Mumar who is Mekadesh the Kidushin is effective, (Shulchan Aruch EH 54:8, 9)