All aircraft tyres will creep, even the slow ones like a Tiger Moth. They creep on initial spin-up on touch-down. If the tyres have inner-tubes then a 'creep mark' will be painted on the wheel and tyre so that the creep can be monitored. Generally the creep mark is about one inch in legnth along the wheel rim, if the tyre creeps by half this amount then the tyre will be deflated and the wheel re-set. If this is not done the wheel can sheer off the inflation valve.
Larger aircraft generally have tubeless tyres and they can creep as much as they like, so they don't have creep marks since there is no danger of the valve being sheered off.
I have never known a wheel to creep by applying full power with the brakes on. When the tyre is under a dynamic load it will grip the wheel more tightly due to its fit on the rim. It is more likely to creep under staic conditions when the tyre is 'at rest' so to speak. It is in this condition when it hits the runway and thus is more prone to creep.