Only when written 歌 / 唄
The ordinary word for a song or the act/sound of singing. 唄 is especially associated with folk songs and shamisen or traditional singing.
I have known this song since I was a child.
My grandmother softly hummed an old folk song.
Only when written 歌 / 詩
A literary use referring to poetry rather than music. The spelling 詩 with the reading うた is poetic and can be ambiguous because 詩 is usually read し.
His poems are highly regarded as modern poetry.
The phrase 愛の詩 in the poetry collection's title is intended to be read as ai no uta.
She sang a song on stage.
She composed a tanka poem while looking at the moon.
He composed a song for his friend.
歌 is the everyday, general word for a song, while 唱歌 usually points to a simple, school-style or old-fashioned song.
歌 is the general word for a song of any kind, while カンツォーネ specifically means a song in the Italian popular/lyrical tradition.
singing contest; singing match
song parody; new lyrics to an existing melody
humming; crooning
love song; love poem
imperial poem; poem by the emperor or imperial family · poem; song (honorific)
poetry gathering; tanka party; tanka competition
poetry contest
children's song; nursery rhyme
counting song · enumerative form (of waka)
break into song; burst into song; begin singing
sad song; lonesome singing
original song
Chinese poem; Chinese-style poem
overtone singing; throat singing
court song for religious services and celebrations
Heian court lady's song; kouta (accompanying ōuta) · popular short song; ditty; ballad (Muromachi to early Edo) · kyogen kouta; love ballad in kyogen · noh kouta; unusual noh song style
Azuma-uta; old Kanto-area folk songs
tanka poet; skilled tanka poet · court poetry official (ritsuryō) · poet (general)
flute mouthpiece; blowhole · poetic style; manner of singing
Attested spelling for poetry, but 詩 alone is more commonly read し; the reading うた is literary or title-like.