I hate hate hate standing martingales, as I feel (like tie downs) all they do is force the horse to not lift their head above a certain height, and unless the horse has a rearing problem you are working to correct and it is on for safety reasons, it's not doing anything right. All it does is give them something to brace on, lean on, and so on.
That said - I do use a tie down on my trail horse bc he HATES bugs and has a habit (even when bug sprayed, fly mask, etc. tried it all) of tossing his head and getting the reins both on one side of his neck. I really think he does it for fun and makes a game of it haha. Anyway the tie down is loose enough it doesn't affect him at all other than when he tosses his head - he doesn't even try to toss his head with the tie on.
As for running martingales, I use them when hunting aggressively, esp on my hot hot hot OTTB who can get very strong. 99% of the time he doesn't need it, but when he throws his head up it's nice to have it there. Bc the running martingale doesn't interfere with the bit or headset at all other than when he throws his head high if excitement, it's there just for safety and not at all for training.
I disagree with using any tie down or martingale (standing or running) for training purposes. I have used with success a german martingale that when adjusted properly has less impact than draw reins and more give than side reins to help rehabilitate a horse that was rebuilding topline. In that case, the german martingale was not used EVERY ride (just like draws should not be used every ride) and was not used to set the head or force the position of the horse, but rather to encourage the head down and stretch through back and hind. Because they were set relatively loose, they were encouraging, not restricting - there IS a very big difference.
Hope that helps!