When is it allowed for a Muslim to raise his hands for supplication? متى يسن للمسلم أن يرفع يديه للدعاء؟

Praise be to Allah, and may Allah’s peace and blessings be upon the Messenger of Allah, his family and his companions.
As to what follows:
Raising of hands is part of the general practice when supplicating. In relation to this, many hadiths have been mentioned. However, raising of hands during Friday sermon is not permissible for the one giving sermon when he is supplicating but he should point with his forefinger only. This is according to a hadith narrated by Muslim that:
Umara Ibn Ruwaiba said he saw Bishr Ibn Marwan on the pulpit raising his hands and said:
 “Allah, disfigure these hands! I have seen Allah's Messenger (peace be on him) gesture no more than this with his hands, and he pointed with his forefinger.”
This view is held by majority of scholars from Maliki, Shafi and Hanbali schools of thought and others too.
As for people other than the khatib, they are regarded as followers and are allweod to raise their hands during supplication. Bukhari narrates a hadith on the authority of Anas (may Allah be pleased with him) who said:

“So, Allah's Messenger (peace be on him) raised both his hands invoking Allah (for rain) and the people too raised their hands with Allah's Messenger (peace be on him) invoking Allah (for rain).”
However, raising hands after obligatory prayers is an innovation because that time is meant for dhikr. The time for supplication is before taslim.
And Allah knows best.

Your brother,
Dr. Khalid al-Muslih
11 / 11 / 1424 AH