Bowing by Modim Derabanan:[1]

When the Chazan reaches Modim the congregation is required to bow together with him in the [beginning of the] recital of Modim Derabanan. Some Poskim[2] rule one is required to bow also towards the end of Modim, upon reciting the words “Al Sheanu Modim Lach.[3]” Practically, it is proper to suspect for this opinion.[4]

The custom:[5] In these provinces [i.e. Ashkenazi Jewry], the custom is to recite the entire Modim Derabanan in a bowing position, from beginning to end.[6] [However some Ashkenazi communities are accustomed to bow only in the beginning of Modim.[7] The  Rebbe was accustomed to bow throughout the recital of Modim.[8]]

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[1] Admur 127/1; Michaber 127/1

[2]  Raavad, brought in Rashba Brachos 34b in interpretation of Rava Brachos ibid; Brought in Admur and Michaber ibid as “Yeish Mi Shemoer”

Other opinions: Some Poskim rule one is to only bow in the beginning of Modim. [Gr”a, brought in Biur Halacha 127 “Hakol”] The custom of the words is only to bow by the beginning of Modim. [Beis Yosef 127; Aruch Hashulchan 127] According to Kabala, one only bows at the beginning of Modim. [Omited in Arizal; Shalmei Tzibur p. 139; Yifei Laleiv 127/3; Kaf Hachaim 127/10]

[3] Admur ibid; Rivivaos Efraim 7/26 that so was custom of Rav Moshe Feinstien; See Piskeiy Teshuvos 127 footnote 14

The reason: As these words contain the Lashon of Modim. Likewise, one is not to bow by “Baruch Keil Hahodaos” being that it is not part of the Nussach according to all opinions. [Rivivaos Efraim ibid]

Other opinions: Some Poskim rule one is to bow by the words “Baruch Keil Hahodaos.” [Chayeh Adam 30/10]

[4] Admur ibid; Michaber ibid; Beis Yosef 127 that so is custom of Midakdikin

[5] Admur ibid; Rama 127/1; Terumos Hadeshen 2/100 [Piskeiy Mahariy-Rav Yisrael Isserlin, 1400’s, Grand student of Mahram Merothenberg]; Opinion in Beis Yosef 127; M”A 127/2 “The custom of the world is like the Rama”; Cutsom of Divrei Chaim, brought in Otzer Hachaim 74

Other opinions: Some Poskim rule one is to straighten his bow when he reaches Hashem’s name and then re-bow at the conclusion of Modim. [Bach 127 in name of Rashal, brought in M”A 127/2; Ateres Zekeinim ibid; Makor Chaim of Chavos Yair 127/1; Custom of Chazon Ish and Steipler, brought in Piskeiy Teshuvos 127 footnote 15]

[6] The reason one is not required to straighten by Hashem’s name: Although in Modim Derabana we mention Hashem’s name of which the ruling is that when Hashem’s name is mentioned one must straighten his bow, in light of the verse “Hashem Zokeif Kefufim”, nevertheless this only applies in the beginning or end of a blessing, and not in its middle. See Admur 113/5. [Admur ibid; M”A ibid; Terumos Hadeshen ibid]

[7] Aruch Hashulchan 127 “By us not all are accustomed to bow”

[8] Hiskashrus 944 footnote 3; However see Hiskashrus 955 that the Rebbe bowed his head numerous times throughout Modim