What are the connotations of "Ali Be'er Enu Lah"?
Rashi: To the well they sang 'Come up well, from the valley and bring up what you found there!' '
Da'as Zekenim: Sing loudly to the well!
Why did they only sing the praises of the miraculous well now, and not during the forty years that it accompanied them on the merit of Miriam?
Rashi: Because of the supernatural miracle that Hashem had just performed through it. 1
As opposed to the ongoing presence of the roving well, which was more of a natural miracle.
Why are the names of Hashem and Moshe not mentioned in Shiras ha'Be'er?
Rashi (on Pasuk 20): Moshe is not mentioned, because he was punished due to it. And since Moshe's name is not mentioned, Hakadosh-Baruch-Hu's Name is not mentioned either. 1
Da'as Zekenim, Moshav Zekenim: They sung the Shirah at the beginning of the 40 years, when they feared lest they die of thirst, just it is written here. Moshe did not sing, because he saw with Ru'ach ha'Kodesh that he will be punished due to it.
Michtav me'Eliyahu (2, p.251): At Shiras ha'Yam, Moshe saw Hashem's Chesed with total clarity (Aspaklaryah ha'Me'irah), and through him, also Yisrael merited a clear vision