Running - for the same reasons as katie_southwest. Beastie sometimes has head flinging moments (for no apparent reason - I've spent good money getting ponio checked out) so I make it into a standing for a while.
Running - TBx likes to toss her head when she strops and it makes a good neck strap. I wouldn't use a standing one incase I didn't use it properly. I've seen too many horses and ponies with huge muscles under their necks and back problems because standing martingales were used incorrectly.
I worked in a saddlery and was told that a running martingale doesnt do anything other than ensure that the contact direction is consistent and that standing is for a high head carriage beyond which there is control.
I am sure there are other theories out there, but its what I have been told.
Don't use either at the moment, but have used both in the past. Standing is a good idea if you think the horse might at some stage fling up its head and bash you in the face. The martingale shouldn't come into action until the horse's head is high, it is not for tying the head down!
Running is quite useful for jumping/cross country as it might stop a horse getting its head up at a crutial point, at take-off going into a fence, but again should only be adjusted so it is a preventative.