What are the connotations of 'Shirah'?
Targum Onkelos and Targum Yonasan: Shirah has connotations of thanks and praise (singing the praises of).
Why does the Torah write "Yashir" (in the future)?
Rashi #2 (citing the Mechilta): It refers to after Techi'as ha'Meisim. 4
Ramban: It is common for the Pasuk to speak in the future even when it is referring to the past. 5
Because the 'Yud' represents Machshavah (Rashi).
On other occasions, the person concerned had in mind to do something, and did not put his thoughts into practice - see for example, Melachim 1, 11:7 (Rashi citing Sanhedrin, 91b). The Ramban queries Rashi from a number of sources - See for example, Tehilim, 106:19, 108:14 and from that entire chapter.
Rashi: This is one of the places where Techiyas ha'Meisim is hinted in T'nach. (See the opening Sugya in the last Perek of Sanhedrin.) Hadar Zekenim - everyone who sings Shirah in this world, he is assured to sing Shirah in the world to come.
Why does the Torah use the double Lashon "Va'yomru Leimor"?
Or ha'Chayim: Each one prompted the other to sing, until they all sang in unison. 1
Which is why "Ashirah la'Hashem" is in the singular (Or ha'Chayim).
What does "Ki Ga'oh Ga'ah" mean?
Rashi #2: Doubled expressions refer to something that nobody else could possibly emulate. When one fights against a man riding a horse, he can only defeat him by toppling him from the horse, whereas Hashem cast the horse together with its rider into the sea.
Rashi #3: However much one praises Hashem, there are always more praises to add - in stark contrast to a human king, whose praises are undeserved.
Seforno: Hashem is deserving of praise for the goodness that He possesses, and not Par'oh, who claimed that the Nile River was his and that he made himself.
Rashbam: The term 'Ge'us' is often used in connection with victory in war - as it does in this case.
Ramban: The horse is proud (vain) in battle, and the rider is proud over the horse (when he rides on its back), and Hashem was proud over both when He cast them together into the sea.
Hashem does not tolerate people who are conceited before Him, and raises Himself above them and punishes them. Consequently, when Par'oh, in his conceit, chased after Yisrael, He cast his horses together with their riders into the Yam-Suf and drowned them (Targum Yonasan).
Why does the Torah here write "Ramah va'Yam" (which implies throwing upwards) and in Pasuk 4, "Yarah va'Yam" (which means He cast down)?
Rashi: This teaches us that the riders together with their horses were cast from the depths of the sea to the top and then down to the bottom again. 1
Why does the Torah write "Sus v'Rochvo" in the singular?
Why does it say "Az Yashir Moshe"?
Da'as Zekenim #1, Hadar Zekenim #1: Moshe said, I complained to You with Az (me'Az Basi El Par'oh), and I will praise You with Az.
Da'as Zekenim #2, Hadar Zekenim #2: The Gematriya of Az is eight. It alludes to Bris Milah, which is on the eighth day, in the merit of which the sea split - "l'Gozer Yam Suf li'Gzarim." The Targum of va'Yamal is v'Gazar, .