Is it required to say "sallallaahu alayhi wa sallam" when we hear the Prophet's name (Muhammad), or when we hear the word "the Rasool"? Please advice with supporting Hadeeth, if possible.

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. We ask Allaah to exalt his mention as well as that of his family and all his companions.

It is confirmed that the Prophet said: "May his nose be rubbed in dust, i.e. may he be humiliated, whom I was mentioned in his presence and he did not send prayers upon me." [Ahmad and At-Tirmithi]

The Prophet also said: "The miser is the one whom I was mentioned in his presence and he did not send prayers upon me." [At-Tirmithi]

As regards to whether one should send prayers to the Prophet when his name is mentioned or when the expression 'Prophet of Allaah' is heard, then sending prayers upon him is required in either cases. The author of Faydh Al-Qadeer said: 'The author of Al-Ithaaf said: 'This is correct, i.e. sending the prayers upon him, when his name is mentioned, his qualities are mentioned, his nickname is mentioned and when mentioning his miracles.' So if the hearer sends prayers upon him when his name is mentioned, or one of his qualities is mentioned or when one of his miracles is mentioned, then all this is permissible.

With regard to the ruling of sending prayers upon the Prophet then Shaykh Al-Islam Ibn Taymiyyah said: 'The Shaafi'ee and Ahmed Schools are of the view that it is obligatory during the prayer and it is not obligatory elsewhere. In another narration of Abu Haneefah, Maalik and Ahmad consider that it is not obligatory during the prayer. Some of the above scholars are of the opinion that it is obligatory only once in a lifetime and some of them consider that it is obligatory in any gathering where his name is mentioned.'   

Allaah knows best.