There are five Sunan of the Athaan as mentioned by Ibn Al-Qayyim in Zaad Al-Ma’ad.
1. Repeating the Athaan
The person hearing the Athaan repeated what the Mu’adhadhin says except for when he hears:
حَيِّ عَلَى الصَّلَاةِ حَيِّ عَلَى الْفَلَاحِ
“Come to prayer, come to prosperity…”
…instead, he says,
لَا حَوْلَ وَلَا قُوَّةَ إِلَّا بِاللَّهِ
“There is no might, or power, except with Allah.”
The benefit of this Sunnah is that it makes Paradise obligatory, as mentioned in the narration by Muslim.
2. Saying the Athkar:
One hearing the Athaan should say:
The benefit of this Sunnah is that the person’s sins would be forgiven.
3. Sending Salah on the Prophet salAllahu ‘alayhi wa sallam
The Prophet salAllahu ‘alayhi wa sallam said: When you hear the Mu’adhdhin say as he says, then send salah on me for whoever sends salah on me, Allah sends salah on him ten times.
The fullest form is Durood Al-Ibraheemiyah:
The meaning of Allah’s salah on his servant is Allah praising the servant in the highest of gatherings (i.e. of the angels).
4. Asking for Intercession of Prophet Muhammad
After the salah (durood) on the Prophet salAllahu ‘alayhi wa sallam say:
The benefit of this supplication is that whoever says it will be granted the Prophet’s intercession on the Day of Resurrection. (Allahumma ja’alna minhum)
5. Supplicating for Oneself
Supplicate for yourself. Ask Allah subhanahu wa ta’ala for His Bounty, for surely He will respond. the Prophet salAllahu ‘alayhi wa sallam said,
Say as they are saying (i.e. the Mu’adhdhin). When you have finished, ask (supplicate), for it will be given to you.
[Book: More than 1000 Sunan Every Day and Night (Adhkar Min Alf Sunnah Fee Al-Yawm Wa Al-Laylah, Dar-us-Salaam Publications)]