The Gemara says that on Shabbos, the shir shel yom by the tamid shel shachar is Mizmor shir l’yom ha’Shabbos, by Musaf it’s the shir of parshas Haazinu – broken up into six segments such that each week there would be a rotation of which portion to say – and for the tamid shel bein ha’arbayim there was a three week rotation comprised of the two halves of shiras ha’yam and shiras ha’b’eir as the third portion.
Now, why Mizmor shir l’yom Ha’Shabbos is relevant to Shabbos we readily understand. It says l’yom ha’Shabbos! But what about Haazinu, Shiras Ha’Yam, and Shiras Ha’b’eir. What do those have to do with Shabbos? Of course, each person can think for himself, but the following is what would appear to be what one can say about the matter.
Chazal say that even a shifcha by Krias Yam Suf experienced a revelation greater than that of Yechezkel ben Buzi. It was a huge experience of ruach ha’kodesh, similar to that of Maamad Har Sinai, just that whereas Maamad Har Sinai was of a permanent nature, the revelation of Krias Yam Suf was temporary. But temporary is also something! That is a parallel to Shabbos. On Shabbos we are given a neshama yeseirah that enables us to reach greater heights of understanding and wisdom. It is an opportunity – whether one feels it or not – to lift oneself up from the realm of mundane and occupy a completely different realm, a totally different reality. A reality that is mei’ein Olah Ha’Bah; literally. It is mei’ein, like echad mi’shishim, but it is not batel. It is a once-a-week, temporary uplift.
The Gemara in Nedarim darshens the Shiras Ha’b’eir as referring to Torah sheh’b’al peh. The whole shira is all about Torah. Shabbos, as well, is a day which is kulo Torah, as the Medrash says in Tana d’vei Eliyahu.
The shir of parshas Haazinu is all about the acharis ha’yamim. It describes, poetically and prophetically, how Ha’Kadosh Baruch Hu is going to bring about the tachlis and end after the long, difficult, winding history of Klal Yisrael. The Ramban says that all of the events of world history are hinted at in sheishes yemei Breishis. Shabbos represents the tachlis when, as Rav Shimshon Pinkus brings in his seifer Shabbos Malkesah, everything will revert back to the beginning. It is tachlis maaseh breishis, and that is why the shir of parshas Haazinu, which describes how Ha’Kadosh Baruch Hu is going to bring about that tachlis, is appropriate as a shir to be said on Shabbos which represents that tachlis.
(Audio Recording [beginning from 25:45] available here)