Answer
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
It is recommended to say du`a after the adhan. When asked about what to say after the adhan, Prophet Muhammad said: “Ask Allah for forgiveness and well-being in this world and the Hereafter.”
In his famous work, Fiqh As-Sunnah, Sheikh Sayyid Sabiq, elaborates on the subject as follows:
After the Adhan, one should make individual supplications, as that is the time when they will most likely be accepted. Anas (may Allah be pleased with him) reported that the Prophet (peace be upon him) said, “A supplication made between the Adhan and the iqamah is not rejected.”
As to the authenticity of this report, it is related by Abu Dawud, an-Nasa’i, and at-Tirmizhi, who called it hasan sahih, and added “They asked, ‘What should we say, O Messenger of Allah?’ He responded, ‘Ask Allah for forgiveness and well-being in this world and the Hereafter.”
`Abdullah ibn `Amr (may Allah be pleased with him) related that a man said, “O Messenger of Allah (peace be upon him), the callers to prayer get more virtues than us.” He said, “Say what they say and when they finish, ask and it shall be given.” (Related by Abu Dawud with a sahih chain)
On the same subject, Umm Salamah (may Allah be pleased with her), “The Prophet (peace be upon him) taught me to say (after) the sunset call to prayer, ‘O Allah, this is the beginning of Your night and the end of Your day. I have supplicated to You, so forgive me.”
Almighty Allah knows best.