Before Prayer
Adhaan (call to prayer) - No power and might except Allah
On hearing Adhaan, repeat what the Mu'aththin says, except for when he says:
حَيَّ عَلَى الصَّلاَةِ وَحَيَّ عَلَىالْفَلَاحِ, recite this Dua.
Prophet ﷺ said, "O Abdullah ibn Qais, should I not inform you to one of the treasures of paradise?"
I said, "Yes, O messenger of Allah". So, he recited this Dua.
لاَ حَوْلَ وَلاَ قُوَّةَ إِلاَّ بِاللَّهِ
Laa hawla wa laa quwwata 'illaa billaah
There is no might and no power except by Allah.
Al-Bukhari 1/152, Muslim 1/288, Sahih al-Bukhari 6384
Adhaan (call to prayer) - Dua to honour Prophet ﷺ
Recite after adhaan.
Ask dua between adhaan and iqamah as duas' asked at this time is not rejected by Allah.
اللَّهُمَّ رَبَّ هَذِهِ الدَّعْوَةِ التَّامَّةِ وَالصَّلَاةِ الْقَائِمَةِ آتِ مُحَمَّداً الْوَسِيلَةَ وَالْفَضِيلَةَ وَابْعَثْهُ مَقَاماً مَحْمُوداً الَّذِي وَعَدْتَهُ [إِنَّكَ لَا تُخْلِفُ الْمِيعَادَ]
Allaahumma Rabba haathihid-da 'watit-taammati wassalaatil-qaa'imati, 'aati Muhammadanil-waseelata walfadheelata, wab 'ath-hu maqaamam-mahmoodanil-lathee wa'adtahu, ['innaka laa tukhliful-mee'aad]
O Allah , Lord of this perfect call and established prayer. Grant Muhammad the intercession and favor, and raise him to the honored station You have promised him, [verily You do not neglect promises].
Al-Bukhari 1/152, and the addition between brackets is from Al-Bayhaqi 1/410 with a good (Hasan) chain of narration. See 'Abdul-Azlz bin Baz's Tuhfatul-'Akhyar, pg. 38