A polite reminder for the Orthodox clergy: In your speeches this shabbos, please do not tell us that Moshe or the Jewish people "sang Az Yoshir." We know what you mean, but its imprecise. The first words of the song are not "az yashir" but "Ashira l'hashem kee goah goah" and the proper name of the song is "Shirat Hayam"
Other notes on the song
Narrative units in the Bible are frequently bookended with long poems. This song marks the conclusion of the Exodus story, and the beginning of the Wilderness tales.
Ashira l'hashem is consistent with ANE literary convention of making announcements at the beginning of poems. (Alter)
Kee goah goah is a great pun. It means "to be exalted" and is also the word for a sea surge.
horse and rider may be an anachronism: At the time Egypt used chariots, not cavalry.
Azi v'zimras is another pun. Zimrah means both song and power.
Who is like you among the gods. possibly indicates that the Israelites weren't quite yet monotheists when then song was first composed.
You will bring them and plant them in the mountain of Your inheritance, The place, O LORD, which You have made for Your dwelling, The sanctuary, O Lord, which Your hands have established seems like a clear reference to the Temple, and can be understood as either prophesy or evidence of later tampering. Sarna, incidentally, uses this to explain away the problem of the too-large multitude that was said to have participated in the Exodus.
As is well known, (see this and this) (and don't miss serious counterarguments in the comments) nothing that we know about the ancient world, and nothing that archaeologists have found in the Sinai lends even an iota of support to the idea that 600,000 (or 3 million) people left Egypt. Sarna's solution is ingenious. He says that Temple in Jerusalem was the point, or goal of the Exodus. At the time the Temple was built, the population in Israel was about 600,000. Saying that 600,000 left Egypt is a literary way of connecting the Exodus with the Temple, similar to how even the children of immigrants speak of their "American forefathers."