All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
It is prescribed in Islam to say the Salah (prayer for blessing) upon mentioning the Prophet after the Athaan. The Prophet said: “When you hear the crier calling the Athaan, repeat what he says, then pray for blessing for me, for anyone who prays for blessing for me will receive ten blessings from Allaah; then ask Allaah for me to be given the Waseelah, as it is an honor in Paradise that is only for one of Allaah's servants, and I hope that I will be that one. Whoever asks that I be given the Waseelah will be assured of my intercession.” [Muslim]
For more benefit, please refer to Fatwa before Athaan and Iqaamah">116221.
It is also required after the Tashahhud. Fudhaalah ibn ‘Ubayd narrated that while the Prophet was seated, a man entered and performed prayer then said, 'O Allaah, forgive me and have mercy upon me.’ The Prophet said: "You have rushed, O man in prayer. When you have performed prayer, sit and praise Allaah as He is worthy to be praised, then ask Allaah to confer His blessings and salvation upon me, then supplicate Him...’” [At-Tirmithi]
For more benefit, please refer to Fatwa 85656.
Moreover, Muslims are enjoined to ask Allaah to confer blessings and peace upon the Prophet whenever his name is mentioned as underlined in Fatwa 86784. The Prophet said: “The miser is the one in whose presence I am mentioned but he does not invoke blessings upon me.” [At-Tirmithi]
A Muslim is enjoined to invoke blessings upon the Prophet whenever he hears his name. If the worshipper repeats his name while invoking blessings upon him, it is enough to invoke blessings upon him once; there is no need to invoke blessings upon him again in this case. This is evidenced by the fact that his name is repeated in the Tashahhud and invoking blessings upon him is mentioned only once.
It is not enough to invoke blessings upon the Prophet in the heart without uttering it verbally. For more benefit, please refer to Fatwa 84707.
Allaah Knows best.