Are there any innovations which some people fall into when at the grave of the Messenger (Peace and blessings be upon him)?

From the innovations that take place at the grave of the Messenger (Peace and blessings be upon him) is much frequentation.

For example, whenever he enters the Masjid, he goes and says 'Salam' to him and sits near the grave.

The Prophet (Peace and blessings be upon him) said:

"Do not take my grave as a place of (much) frequentation." [Musnad Ahmad, Musannaf 'Abdur-Razzaq]

Rather, it is recommended for one who arrives from a journey to visit it.

Also, from the innovations that occur at the Messenger's grave or other than that is the assumption that the supplication there is accepted.

On the contrary, it is legislated to send salutations upon him (Peace and blessings be upon him), and if he wants to supplicate then it is to be done in any part of the Masjid and if it is done after the prayer then that is best.

From that which is done and disliked at the grave of the Prophet (Peace and blessings be upon him) is raising the voice and requesting from him (Peace and blessings be upon him) that one's needs be satisfied. This is a major shirk, and that which is obligatory is to be cautious of it.