I would guess the majority are just poking fun not a major insult to your pride.  I see it all over (not BYU vs Utah, but other schools).  I have also seen BYU fans with snide comments to Utah fans, it happens in my ward all the time with adults even saying things to the youth (I don't live in Utah so maybe not as offensive).....not a big deal at all just fun ribbing.

I understand it is horrible if it is a threat or bad language, but if someone says "BYU stinks", "how is independence", "great year last year", "its been a tough decade" etc etc that it is not worth getting offended.  They are dumb comments, we all say similar things so don't get in a huff. 

If you go to any school that is relevant anywhere in the US, there is a very good chance a rival fan will make a comment.