sing vs carol

sing is not a noun while carol is a noun.

Word	Noun	Adjective	Verb	Adverb
sing	No	No	Yes	No
carol	Yes	No	Yes	No

As verbs, carol is a hyponym of sing; that is, carol is a word with a more specific, narrower meaning than sing:

Other hyponyms of sing include solmizate, troll, hymn, madrigal.

sing (noun)	carol (noun)
	a joyful song (usually celebrating the birth of Christ)
	joyful religious song celebrating the birth of Christ

sing (verb)	carol (verb)
make a whining, ringing, or whistling sound	sing carols
divulge confidential information or secrets
to make melodious sounds
produce tones with the voice
deliver by singing