Many are accustomed to the fact that in rabbinic Judaism a Jew is one who is born of a Jewish woman.
But were there opposite opinions? Poskim of post-Talmudic time are of also interest.
As pointed out by @Chatzkel (see comment) Kiddushin 68b it appears clear through a number of verses that Judaism is determined by the mother. There does not appear an opinion that argues on this point.
I personally don't recall an opinion that stated that "Jewishness" is decided patrilineally.
In regard to the father:
This question is mostly discussed in the third and fourth perek of Kiddushin. There are issues in regard to lineage (mamzer etc...) that do include Patrilineal descent as well in to the equation. These questions affect who the child as a Jew may marry.
Similary what tribe a person belongs to is decided by patrilineal descent. For example a Kohen or Levi is decided through the father.