Does it say in the tashahhud “al-salaam ‘ala al-nabiy” or “al-salaamu ‘alayka ayyuha’l-nabiy”?

What is prescribed is to say that which the Prophet
(peace and blessings of Allaah be upon him) taught to the Sahaabah. He
taught them to say: “al-salaamu ‘alayka ayyuha’l-nabiyyu wa
rahmat-Allaahi wa barakaatuhu (peace be upon you, O Prophet, and the
mercy of Allaah and His blessings)”. So we say what the Prophet (peace
and blessings of Allaah be upon him) taught us and taught the Sahaabah, and
he did not tell them “When I die, change it.” He taught it to them when they
were travelling and going to far distant lands, and they would say “al-salaamu
‘alayka ayyuha’l-nabiyyu wa rahmat-Allaahi wa barakaatuhu”, i.e., they
were praying for him. Saying salaam to him is praying for him, for well
being, mercy and blessing. “ayyuha’l-nabiy (O Prophet)” is just
referring to him; “ayyuha’l-nabiy (O Prophet)” does not mean calling
him, rather they are praying for him. “Al-salaamu ‘alayka (peace be
upon you)” means may you be safe, may you granted peace, well being, mercy
and blessing from your Lord. It is a du’aa’ or supplication for him
(peace and blessings of Allaah be upon him), not calling upon him; rather
you are asking Allaah to grant him well being, mercy and blessing. If
someone says “al-salaamu ‘ala al-nabiy wa rahmat-Allaahi wa barakaatuhu
(peace be upon the Prophet and the mercy of Allaah and His blessings)” there
is nothing wrong with it, but it is better to say what the Prophet
(peace and blessings of Allaah be upon him) taught the Sahaabah: “al-salaamu
‘alayka ayyuha’l-nabiyyu wa rahmat-Allaahi wa barakaatuhu (peace be upon
you, O Prophet, and the mercy of Allaah and His blessings)”. This is what
the Prophet (peace and blessings of Allaah be upon him) taught to his
ummah, and he died when that was the case, (peace and blessings of
Allaah be upon him). End quote. 

Majmoo’ Fataawa Ibn Baaz
(28/58). 

See also the answer to question no.
34535.

and Allaah knows best.