Which is correct?
I would like to request you to refrain from shouting.
I would like to request of you to refrain from shouting.
I would like to request from you to refrain from shouting.
3 is out: one might request an item from a person, but not generally a behavior.
2 is correct but the "of" is unnecessary.
1 just sounds better than the others because it is less wordy.
However, the most direct is, "Please refrain from shouting." Why make it complicated?
There is another obvious one to add to your list:
I think there are people who use all three forms you suggested. It's a little difficult to say which one is correct but as kajaco said 3. sounds a bit odd.
The simple answer is that it varies. You can:
Request [a person] to [do something]~ 'request visitors to remove their shoes'
Request to have [a thing or a favour]~ request to have his telephone upgraded'
Request for [a thing or favour] ~ 'a request for a new computer'
Request that [something happens] ~ 'request that you refrain from shouting'
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