At the time of the Adhan there is no better dhikr than repeating the words of the Adhan after the muezzin. Once the Adhan is over, we should offer the supplication prescribed for the occasion as has been reported in the sources.
It is reported by Al-Bukhari and Muslim on the authority of Abu Sa`id Al-Khudri that the Messenger of Allah (peace and blessings be upon him) said, “When the muezzin calls to Prayer, you should repeat the words after him.”

Sheikh M. S. Al-Munajjid, a prominent Saudi Muslim lecturer and author, says:

1. With regard to du`aa’ before the Adhan, there is none to be recited before the Adhan as far as I know. If that time is singled out for any type of du`aa’, this is a reprehensible innovation (bid`ah). But if it is done by coincidence and accidentally, then there is nothing wrong with it.
2. With regard to du`aa’ before the Iqamah, when the muezzin is about to start the Iqamah, we know of no specific words to be recited at this time. Doing so when there is no solid evidence in Shari`ah is a reprehensible innovation (bid`ah).
3. With regard to the time between the Adhan and Iqamah, du`aa’ is encouraged at this time and it is mustahabb “Du`aa’ is not rejected between the Adhan and the Iqamah, so make du`aa’” (At-Tirmidhi, Abu Dawud, and Ahmad).

There is a specific du`aa’ to be recited immediately after the Adhan.

It was narrated from Jabir ibn `Abdullah that the Messenger of Allah (peace and blessings be upon him) said, “Whoever says when he hears the call to Prayer, ‘O Allah, the Lord of this perfect Call and the Prayer we are about to offer! Grant Muhammad the privilege (of intercession) and honor, and raise him up to the eminent position that You have promised him,’ my intercession for him will be granted on the Day of Resurrection” (Al-Bukhari).
4. With regard to du`aa’ after the Iqamah, we know of no evidence for that. If any specific du`aa’ is made in the absence of any sound evidence, it is an innovation (bid`ah).
5. With regard to du`aa’ at the time of the Adhan, it is sunnah to repeat what the muezzin is saying, except that when he says “Hayya `ala as-salah, hayya `ala- al-falah (Come to Prayer, come to success),” you should say “Laa hawla wa laa quwwata illa billaah (there is no power and no strength except with Allah).”

It was narrated that `Umar ibn Al-Khattab said: the Messenger of Allah (peace and blessings be upon him) said, “If the muezzin says ‘Allahu akbar, Allahu akbar’ and one of you says ‘Allahu akbar, Allahu akbar’; then he says ‘Ashhadu an laa ilaaha ill-Allah’ and you say ‘Ashhadu an laa ilaaha ill-Allah’; then he says, ‘Ashhadu anna Muham
madan Rasul Allah’ and you say, ‘Ashhadu anna Muhammadan Rasul Allah’; then he says, ‘Hayya `ala-s-salah’ and you say ‘Laa hawla wa laa quwwata illa billah’; then he says, ‘Hayya `ala-l-falah’ and you say ‘Laa hawla wa laa quwwata illa billah’; then he says, ‘Allahu akbar, Allahu akbar’ and you say, ‘Allahu akbar, Allahu akbar’; then he says ‘Laa ilaaha ill-Allah’ and you say, ‘Laa ilaaha ill-Allah’ from the heart, you will enter Paradise” (Muslim).
6. With regard to making du`aa’ at the time of the Iqamah, some of the scholars regarded it as being like the Adhan, so it is mustahabb to repeat the words. Other scholars did not regard it as mustahabb because of the weakness of the hadith that speaks of repeating the words of the Iqamah.
It is a mistake to say “Aqamaha Allah wa adamaha Allah” when the muezzin says “qad qamat as-salah,” because the hadith narrated concerning that is da`if (weak).
It was narrated from Abu Umamah or from one of the Companions of the Prophet (peace and blessings be upon him) that Bilal began to recite the Iqamah and when he said “qad qamat as-salah,” the Prophet (peace and blessings be upon him) said “aqamaha Allah wa adamaha Allah.” And he said concerning the rest of the Iqamah something like what is mentioned in the hadith of `Umar (may Allah be pleased with him) about the Adhan (Abu Dawud). This hadith was classed as da`if (weak) by Al-Hafizh ibn Hajar.