Dear Rabbi Why do some shuls say the shir shel yom after shacharis (before kryas hatoarh) while others say it after musaf?
ב"ה Shalom The question you have asked is more far reaching than what you may have thought when you asked it. I will explain, but at first I can simply answer that it is a difference of customs between Nusach Ashkenaz and Nusach Sefarad. Nusach Ashkenaz chose to say it after Musaf on Shabbat while Nusach Sefarad chose to say it after Shacharit. But the underlying reasons make your question more complex, because the custom to say Ein ka'elokeino and Shir shel yom are brought in the shulchan aruch as prayers to be said at the conclusion of shacharit, remembering the the shir the Leviim said in the Bet Hamikdash after the Korban Tamid. The shir shel yom reminding us of the Nesachim in the Bet Hamikdash (משנה תמיד ז:ד). So on one hand it makes perfect sense to say shir shel yom after Shacharit on Shabbat the same as the weekday. However, according to Nusach Ashkenaz and Nusach Sefarad, nobody says Ein ka'elokeino after Shacharit on Shabbat only after Musaf. This then raises the question of why delay Ein ka'elokeino which is to remind us of the Ktoret which according to some should even be said before Shacharit.(ט"ז סי' קל"ב:ב) . Some then raise also the question of why shir shel yom is said by some before Ein ka'elokeino,which reminds us of the Nesachim, and should not to be said before the Ein ka'elokeino which is to remind us of the Ktoret, since the Ktoret is before the Nesachim. (סידור צלותא דאברהם עמ' שצג ,רמב"ם הלכות תמידים פרק ו,) I have not found an explanation for that, but then some also question the whole reason for saying shir Shel yom which was not mentioned by most of the rishonim and even the Rambam in his order of the Tefilla mentions only that some people have the minhag to say shir Shel yom. It has therefore been suggested that this follows the custom of Kabbalists which has been followed in our later generations. (סידור צלותא דאברהם עמ' שצב). So, I am sorry if I have not fully answered your question and only raised more for you and myself. All the best and Besorot tovot to you and all of Am Yisra'el.