It seems that not everyone [always] sang/sings Aishes Chayil with Sholom Aleichem at the Friday night Shabbos table.
When and how did the custom of singing Aishes Chayil at the table begin? Who follows it--or rather, who doesn't?
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The custom of saying Eshet Hayil on Friday night does not seem universal among Teimani Jews. R. Qafih makes no mention of it in his discussion of Friday night practices in Yemen (Halikhot Teiman (1969) pg. 5). The Teimani Siddur Siah Yerushalayim (pg. 48) states that some add Eshet Hayil.
Based (partially) this article, the practice was popularised by the work Tikkunei Shabbat (373) presenting the ideas of the Arizal (mid 16th cent.). It is in turn quoted verbatim in the controversial but influential Hemdat Yamim (17th cent.) here. It was further referenced by Shelah (late 16th-17th century) over here, R. Yaakov Emden (18th cent.) here, R. Zaddok Hakohen of Lublin (mid 19th cent.) (P'ri Tsadik: Parashat Hayei Sarah: 1), Ben Ish Hai (e.g. Rav P'alim (Vol. IV: Sod Yesharim: 16) (late 19th cent.) and many others.