In the old days, in the homes of an earlier generation of Catholics, I remember them being displayed behind sacred art, paintings of the Sacred Heart, the sorts of iconography I don't have.
But in the last 10 years or so, my church has put out a call for old palms a few weeks before Lent begins. They burn the palms and use the ashes on Ash Wednesday — a lovely example of religious recycling.
I was able to shed a large backlog of palms that way. Now, my house is almost palm free. The "almost" is because ... I picked up another palm yesterday.