I see many people raising their hands when reciting the dua after athan. I have also seen some scholars not raising their hands? What is the correct procedure?
In the Name of Allah, the Most Gracious, the Most Merciful.
As-salāmu ‘alaykum wa-rahmatullāhi wa-barakātuh.
In essence there are two types of du῾ā’s:
1) Those which do not consist of any particular wording for any given situation. For example, a person makes his own du῾ā’ for a need he has after performing two rak῾at of voluntary prayer.
2) Those du῾ā which have specific wordings and phrases at particular times and instances. For example, the du῾ā’s after eating, after sleeping etc.
The Ahādith pertaining to raising the hands for du῾ā’ are in relation to the first category and not the second category. The second category of du῾ā’s are considered as athkār (remembrance of Allah) and thus, the hands should not be raised for such utterances.
The du῾ā’ after adhān falls under category two, therefore, the hands should not be raised when reciting the du῾ā’ after adhān.
And Allah Ta’āla Knows Best
Mufti Faraz ibn Adam al-Mahmudi,
 فتاوى دار العلوم زكريا ج 2 ص 92 زمزم
This answer was collected from DarulFiqh.com, which is operated under the supervision of Mufti Faraz ibn Adam al-Mahmudi, the student of world renowned Mufti Ebrahim Desai (Hafizahullah).