Shaĥarit was instituted to correspond to the morning Tamid offering, and after the offering of the Tamid sacrifice in the Temple, the Levites would recite the Shir shel Yom (Psalm of the Day). It therefore became customary to recite the Shir shel Yom after Shaĥarit (Sofrim 18:1). Although it was not instituted originally as an obligation and some did not recite it, by the end of the era of the Rishonim, all had already adopted it.
Before reciting the psalm, there day is invoked according to the format: “Today is the first day from Shabbat,” in order to fulfill the mitzva of remembering Shabbat every day of the week (based on Ramban on Shemot 20:8; Arizal as cited in Kaf Ha-ĥayim 132:26).
After the Shir shel Yom, it is customary to recite Pitum Ha-ketoret, introduced by the liturgical poem Ein K-Elokeinu. There are two reasons for the recitation of Pitum Ha-ketoret: The first is that it corresponds to the incense which was offered every morning and evening in the Temple. The second is so that all men, who are commanded to study Torah, may study the words of the Sages each day, and additional aggadic (non-legal) materials were appended to ensure that men study aggada each day as well (see Peninei Halakha: Prayer, 23:5 n. 5).
Aleinu is regularly recited at the conclusion of Shaĥarit in order to imbue faith in God and hope of redemption in our hearts, and so that later, when the Jew who prayed meets non-Jews at work and through business, he will not be enticed by their gods and beliefs (Baĥ §133). Based on Arizal, it became customary to recite Aleinu after Minĥa and Ma’ariv as well. Due to the significance of this prayer, it is customarily recited standing, and one bows slightly when saying the words “Va-anaĥnu kor’im” (“We bow”) in the Ashkenazic rite and “Va-anaĥnu mishtaĥavim” (“We prostrate ourselves”) in the Sephardic rite (MB 132:9).
Since these passages are all relatively late additions to the prayer service, there are differences between various versions of them. For instance, in the Sephardic rite, more songs and verses are added before the Shir shel Yom and the wording of Pitum Ha-ketoret is longer. There is also a difference in the order of the prayers: in the Ashkenazic rite, Aleinu is recited before the Shir shel Yom, whereas in the Sephardic and Ĥasidic rites, Shir shel Yom is recited first, then Pitum Ha-ketoret, and finally, Aleinu (see Peninei Halakha: Prayer 23:4-5).