It is makrooh to sing the adhaan in the sense of elongating the words excessively without changing the meaning. If the meaning is changed then it becomes haraam and the adhaan is not correct in this case.
Ibn Abi Shaybah (259) narrated that a muezzin gave the adhaan and elongated the words in his adhaan and made it sound like singing. ‘Umar ibn ‘Abd al-‘Azeez said to him: Give a gentle adhaan otherwise keep away from us.
What is meant by a gentle adhaan is one without any singing it.
See al-Maghrib fi Lughat al-Fiqh by Abu’l-Makaarim al-Matrazi al-Hanafi (p 234)
It says in al-Mudawwanah (1/159):
Maalik regarded it as makrooh to sing the adhaan and disliked that intensely.
Imam al-Shaafa’i said in al-Umm (1/107):
I like the adhaan to be recited clearly without elongating any words or rushing it.
It says in al-Mawsoo’ah al-Fiqhiyyah:
The fuqaha’ are agreed that elongating the words of the adhaan and singing it and adding any elongation or letter at the beginning or the end is makrooh, because it is contrary to proper focus and dignity. But if the singing and elongation changes the meaning then it is haraam and there is no scholarly dispute on this point, because it was narrated that a man said to Ibn ‘Umar: “I love you for the sake of Allaah.” He said: “I hate you for the sake of Allaah, for you sing the adhaan.”
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said, concerning adhaan that is sung, this means giving the adhaan with a tune as if singing a song. It is valid but it is makrooh.
And he said: But making grammatical mistakesi n the adhaan falls into two categories, one which renders the adhaan invalid, which is when the meaning is changed, and one in which the adhaan is valid but is makrooh, which is when the meaning is not changed. For example, if the muezzin says “Allaahu akbaar” (elongating the ‘a’ sound), this is not valid because it changes the meaning, because akbaar is the plural of kabar, which is a type of drum.
From al-Sharh al-Mumti’, 2/62
Shaykh Muhammad ibn Ibraaheem (may Allaah have mercy on him) said: Excessive elongation is not appropriate, and if the vowels are elongated to such an extent that the meaning is changed, this is not valid, otherwise it is makrooh. From Fataawa al-Shaykh, 1/125.
The point is that singing the adhaan is makrooh, and may render the adhaan invalid if it changes the meaning.
And Allaah knows best.