Are we required to bow during the 'Aleinu prayer when reciting the line starting with ״ואנחנו כורעים״?
If there are differing halakhic rulings or customs, which I assume there are, their sources would be much appreciated!
While I've yet to see a source that says that one shouldn't bow, sources that say that one should include the following (in rough chronological order):
• Emek Berakha (§40, here), by R' Avraham ben Shabbetai Horowitz, with notes by his son, the Shnei Luchot haBrit. No reason is provided for the bowing, save that the whole host of heaven is listening.
• Magen Avraham, OC 132:2, s.v. כורעים. He gives the reason that one shouldn't appear to be a heretic by not bowing. He sources his opinion in the Emek Berakha.
• Acc. to Siddur Vilna, R' Yaakov Emden mentions this custom in his siddur (לוח ארש), although I've not checked the language.
• Be'er Heitev (R' Yehuda Ashkenazi), OC 132:4, s.v. ואנחנו כורעים. He doesn't provide a reason, but he references the Arizal as well as the Magen Avraham.
• Arukh haShulchan 133:1. Like the Be'er Heitev, R' Epstein also doesn't provide a reason, and he also references the Arizal.
• Mishna Berurah 132:9. Although the MB doesn't cite a reference, the reasoning he provides is the same as the Magen Avraham's: so that one not appear to be a heretic.
This is a custom that the Rema doesn't mention at all (although he does say that one should pause briefly before reciting those words; OC 132:2), and perhaps the fact that he doesn't mention it is noteworthy (?). Likewise noteworthy might be the fact that the Qitzur SA does mention this tradition when it comes to Rosh haShana - despite asserting that we don't fall on our faces then as we do on Yom Kippur (129:16), but doesn't mention it at all when discussing the regular weekday/Shabbat Aleynu (25:6).
According to the Art Scroll Siddur bend the knees at kor'im (bend the knee), bend at the waist at mishtachavim (bow) and straighten up after modim (acknowledge) before lifnei melelech.
Of course this is just the instruction posted in the siddur. I do not know where Rabbi Nosson Scherman obtained the particular psak that this is the correct methodology. I do remember being taught this in cheder, but that also was part of learning how to daven and I no longer remember the source.