Putting powder (or resin/rosin) INSIDE the thumb or finger holes is acceptable to help to keep from sticking in the ball, but it can't be so much that powder/rosin comes out of these holes after the ball is released. Putting powder (or resin/rosin) on the outside of the surface of the ball is prohibited. These are rules for any U.S. Bowling Congress sanctioned play, and probably would apply to any other competition, too. As far as open play goes, what you can get away is what you can get away with, but most (or nearly all) bowling centers wouldn't allow you to get powder all over the place, the same way they wouldn't let you track water or mud all over. As far as powder on SHOES, that isn't allowed, though it is often done and not looked at if done in a way to not leave a residue anywhere.
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No. In league play, foreign substances are prohibited from being applied to the soles of the shoe.
Current USBC rules prohibit foreign substances on your shoes during league or tournament play. This substance could possibly come off n the approach area and be picked up by other bowlers.