From what I have seen by attending services as well as viewing various Siddurim, it seems that on weekdays, the Song of the Day is said at the end of Shacharit (Nusach Ashkenaz after Aleinu; Nusach Sefard before it.)
However, on Shabbat and Yom Tov, I have seen it placed in one of 3 places for both Nusach Ashkenaz and Sefard:
- Beginning of service just before start of Psukei Dezimra
- Just after Chazzan's Shacharit Amida repetition
- End of Musaf service either before or after Shir Hakvod or Aleinu (Shir Hakavod is also "portable".)
Why is there this flexibility on Shabbat / Yom Tov but not during the weekday?
Speculation: It's either the day itself or the fact that we say Musaph that creates the ambiguity.