I've heard that one should undo a standing martingale while crossing a river in case the horse falls and finds himself/herself in the dangerous situation of having to get up in a river! Their head and neck cannot extend out fully with a standing martingale attached. So, if the river's flowing and high, take the time to undo that standing martingale.
I have not witnessed this but just some advice for any of us who have to use those awful standing martingales while hacking.
Also, something I do for safety's sake...I put a string around the top neck loop of my standing martingale and attach it to the ring on the front of the saddle with a little snap. This keeps the loop up where it should be in case I'm grazing her on the trail and it slips down.
Happy Trails everyone!