What is the ruling for invoking blessings upon the Prophet after the adhan?

Invoking blessings upon the Prophet after the adhan is a sunnah
Invoking blessings upon the Prophet after the adhan [call to prayer] is firmly established in the corpus of authentic hadiths. 'Abdullah Ibn 'Amr Ibn al-'As (may Allah be pleased with them both) narrated that he heard the Prophet say, "When you hear the mu`adhin calling for prayer, repeat what he says and then invoke blessings upon me, because whoever does so will be earn a tenfold reward. Then beseech Allah to grant me al-Wasilah: it is a high rank in Paradise fit for only one of Allah's servants, and I hope to be that person. I will intercede for whoever asks that I attain al-Wasilah" [Recorded by Muslim, Abu-Dawud, at-Tirmdhi and an-Nasa`i].

The manner of invoking blessings upon the Prophet after the adhan There is no text that obligates a single manner, audible or silent, for invoking blessings upon the Prophet;the matter is open and capacious. When Allah Almighty legislates a matter in a general way and it may be executed and applied through various means, it must be left general and it is impermissible to restrict it to a certain manner without supporting proof.

There are reports that evidence the [permissibility of] invoking blessings upon the Prophet audibly after the adhan. Abu al-Darda` (may Allah be pleased with him)narrated that after hearing the adhan, the Messenger of Allahwould say, "Allahuma Rab hadhihi ad-da'wah at-tama was-salah al-qa`imah, salli 'ala Mohammed wa a'tih su`alihu yawm al-qiyamah [O Allah, Lord of this most perfect call and prayer that is now being established, send your blessings upon Muhammad and grant him his wish on the Day of Resurrection]" [Recorded by At-Tabarani in Al-Mu'jam Al-Kabeer]. This was heard by those around him who would repeat themu`adhin'swords. The Prophet said, "Whoever says this when he hears the mu`adhin, it becomes incumbent upon me to intercede for him on the Day of Judgment."

The ruling

The matter is open and the right thing to do is to let people behave naturally. Whoever wants to send blessings upon the Prophet is free to make it in the manner and words which best appeal to him since it is important that the manner and formula resonate in his heart. It is impermissible for anyone to deny what others do as long as the matter is open and capacious.

Allah Most High knows best.