As far as poles are concerned, straps off when you ski anywhere it might get snagged (trees, urban, chairlift, etc). Personally i never wear pole straps unless I'm doing big lines. Even then sometimes I don't wear them but that's me personally, I have a death grip when I fall.
The main reason for not wearing them while tree skiing is not actually the danger of getting it snagged and hurting your shoulder. That's a common misconception. The main reason for not wearing pole straps in the forest are tree wells. Often in tree wells, snow might fall off from a branch above you. If you get stuck in a tree well, you poles act as anchors pinning your hands down and its actually really dangerous due to suffocation (especially if you fall in head first).
This can also apply to skiing and falling in a creek with snow falling on you or in my case, a pillow line. I got buried tits down by a pillow that lost its snow above me and although I was fine. My poles had anchored my buried arms and I needed a friends help to get out.