Pretty darn hands-off here. One exception is that with a large baby and either a first time vag birther and/or a perineum that seems real tight, I will put the tiniest amount of downward pressure with a fingertip on the baby's head right below mom's clit (unless she is reaching there herself, some will, some won't). This is to keep any possible tearing from going upward. Last time I did this, mom did have a tiny flesh wound down low--no problems with it, only a little pain for a few days. Can't help but think how much more it would have hurt, and impacted her sex life, if that tear had been at the clit instead, as it seemed likely to do before I put my finger there.
All that said, like mwherbs I encourage moms to be in hand contact w/baby as it emerges, if not actually catching; I also encourage dads to catch but it seems often that the moms want dads otherwise holding onto her. And also, there are some moms who really want my hands on the perineum, and/or massaging just on the inside--they find that this helps key them to relaxing the area maximally as babyburn happens. If my hands are on outside, then it's as little pressure as I can manage and still be touching them--it's NOT counterpressure, just a 'touch relaxation' thing. If on the inside, it's a gentle but firmish massage pressure, moving around the circle slowly.