... or why I find it so very difficult, and try to keep out of it!
A very good explanation of what the sign of peace at Mass is really about, and the difficulties that its practice presents, can be found here. Please do not get me wrong. I am actually very keen on the sign of peace, and keen on it staying where it is, as part of the Communion rite (there are suggestions of moving it to an earlier point in the liturgy, at the Penitential rite or the Offertory). But I just can't cope with it the way it is usually celebrated at Mass these days.
For just over a year now, after a couple of exceptionally bad experiences, I have tried to keep out of exchanging the sign of peace at Mass (though courtesy doesn't always make it possible). The worst case scenario occurs when those around you demonstrate little sense of proper participation at Mass (yes, I know, judgemental, and I wouldn't express this at an individual level) and then enthusiastically expect you to shake hands with them! Another scenario occurs when those around you do participate with reverence and devotion (just as judgemental as above, I know) and then offer the sign of peace in terms which demonstrate a complete lack of understanding of what it is meant to be. The problem here, at least in part, is that the handshake that is the defined sign to be exchanged in England and Wales is not a sacred sign at all but a rather every day sign. I think there is some mileage in the practice I experienced at one time of omitting the exchange of the sign between the people (note that this does not omit the offering of the peace from the celebrating priest to the people, which represents the essence of the sign of peace in any case) during the week, but including it at Sunday Masses.
The classic occurred this morning when the very devout people around me at Mass (a splendid turn out on a Saturday morning for a New Year's Day Mass, including a good number of families) ..... offered the sign of peace wishing me a happy new year!