A most interesting blog on the sad state of parochial music in Roman parishes by the always worth your time Kathy Pluth suggests that it is an odd occurrence for random parishioners in the pew to start up whatever hymn they choose during Mass.
I hope that that is so, that it is indeed rare, although I have lived in two different parishes, (in the US,) where this was liable to happen.
Although in one of them, I guess it was not so very random, it would only ever be O Lord I Am Not Worthy, God Bless America or Let There be Cheese in Church.
I was emboldened to start, on my own... motu proprio? humming the Gregorian Ave Verum once during communion after OLIANW's three verses were finished....
Felt bad about it afterwards.