Whose name do you mention-the father or the mother:[1]
Mi Shebeirach/Praying for the sick:[2] When praying for the sick one is to mention his/her name and the mothers name. Thus, when saying Mi Shebeirach for the sick, one is to mention the mother’s name.
Mi Shebeirach Leyoledes:[3] By the Mi Shebeirach for a Yoledes one is to mention the mother’s name of the child.
Yizkor:[4] During Yizkor one is to mention the mother’s name of the deceased.
Mishnayos:[5] When learning Mishnayos in memory of one who is deceased, one is to mention the name of the father.
Matzeiva:[6] On a Matzeiva one is to write the father’s name of the deceased.
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[1] See Ziv Hasheimos chapter 29
[2] See Sefer Chassidim 237; Zohar Lech Lecha p. 84; Shulchan Menachem 1/266
[3] Siddur Tehilas Hashem
[4] Chabad custom
[5] Sefer Ziv Hasheimos ibid
[6] Sefer Ziv Hasheimos ibid