Amphibious seaplanes can have shimmy problems with the nose gear - and like the tail draggers when they do the nose wheels do 360s around the castering pins.
Non engineering factors that seem to encourage shimmy include:
Higher groundspeeds (light or or tail wind, heavy loads)
Hard surface (never had a noticeable shimmy on grass or loose gravel)
Dropping the nosewheels onto the runway (rather than easing them down)
Rough runways
Forward Centre of Gravity
If you can't lift the nose gear back into the air, a significant reduction in speed is often the only thing that will stop a bad shimmy. Sometimes unloading the nose gear can even make the shimmy worse. Heavy braking is often the only option - and all it does is reduce the number of oscillations by reducing the time that the wheel shimmys.
I've seen a bent trolley axle and destroyed trolley blocks after only four landings after new trolley blocks had been fitted. Of those four landings it was only the last that experienced shimmy - and apparently it was so violent it nearly pulled the aircraft off the runway.