What is the ruling on shaking hands after the prescribed prayer in the mosque, especially during Eid days (one day after Eid)?

Shaking hands after the five daily prayers is not supported by any evidence for its permissibility; in fact, some scholars have deemed it reprehensible. However, shaking hands on Eid day, if it is for the purpose of meeting between two people, is a Sunnah. As for congratulating someone on Eid and at the beginning of months, that is a Sunnah, but it does not necessitate the desirability of shaking hands.