The Shacharit prayer was instituted to correspond to the morning Tamid offering, and after the sacrifice of the Tamid, the Levi’im would recite Shir Shel Yom (Psalm of the Day). Therefore, it became customary to say Shir Shel Yom after the Shacharit prayer service (Masechet Sofrim 18:1). Although it was not instituted originally as an obligation, and some did not regularly recite it, by the end of the period of the Rishonim, everyone had already become accustomed to doing so.
Before the Psalm, there is reference to the day: “Today is the first day of the week in relation to Shabbat,” in order to fulfill the mitzvah of remembering the Shabbat on every day of the week (based on Ramban Exodus 20:8; the Ari as brought in Kaf HaChaim 132:26).
In Nusach Sephard, the Psalm “Tefillah L’David” (Psalm 86) is added before Shir Shel Yom, as well as a collection of verses which start with “Beit Yaakov” and “Shir HaMa’alot L’David” (Psalm 124). On days of joyous occasions, when we do not recite Lamenatze’ach mizmor l’David, ya’ancha Hashem b’yom tzarah (after Ashrei), Tefillah L’David is not said either, since the words “b’yom tzarati ekra’eka” (“on the day of my trouble I call You”) are mentioned in it (Piskei Teshuvot 132:11). These passages are also printed in the siddurim of Nusach Sephard-Chassidi, although many omit them, only reciting Shir Shel Yom, like Minhag Ashkenaz.
Another difference between the nusachim is that in Nusach Ashkenaz, Aleinu L’Shabe’ach is recited first, then Shir Shel Yom, and finally Pitum HaKetoret; whereas in Nusach Sephard, Shir Shel Yom is recited first, then Pitum HaKetoret, and at the end, Aleinu L’Shabe’ach (the order of discussion in this book is arranged according to the order of prayers in Nusach Sephard).
Concerning someone who practices according to one nusach and is praying with a chazan following a different nusach, there is disagreement. Some say it is best that he prays in the nusach of the chazan, and others say it is best that he prays quietly in his own family’s nusach, but that he may not sit when the whole congregation stands for Aleinu L’Shabe’ach, so as not to emphasize the differences (see also earlier in this book, 6:5).
. Siddur Rav Amram Gaon mentions that Shir Shel Yom is recited. The Rambam, at the end of Sefer Ahavah in his wording of prayer, writes, “Some Jews were accustomed to reciting it…” In the Temple, Shir Shel Yom was also recited after the afternoon Tamid, nevertheless, our custom is not to recite it at Minchah. The reason for this, explains the Mishnah Berurah 132:16, that at times, in the Temple as well, if they were late in bringing the libation, Shir Shel Yom was not recited, because songs of praise are not recited at night.