The Church desires that Catholics should read the bible. She further points out that the bible can only be correctly understood in the light of the Church - that is, in the light of the teaching and Tradition of the Church. To simply pick up the bible, start reading, and assume one can understand it without error, is a guaranteed way to enter into error. Remember, every heresy uses the bible for support.
Catholics typically don’t bring bibles to Mass because at Mass there is never the type of bouncing back and forth between passages that happens in many Protestant services. Selected passages of scripture are read on each given day all throughout the Church (one sign of her universality). These readings are on a three-year cycle, and most churches will have missalettes (sp?) which contain the readings, psalms, responses, etc for a given day. In addition, many people bring their own missals which have the same information in a more permanent format.
BTW, nobody would ever be frowned upon if they did bring a bible. It’s just that there is no need, and poking around in the bible outside of the scripture readings could easily distract from the rest of the Mass, where we are in the very real presence of God. As important as the scriptures are, they are not the pinnacle of the Mass.