Behaviour of water at exactly 0 °C
It depends on how you achieved water at 0 °C. If you cooled water down to 0 °C, then it will remain water, unless you remove more heat, at which point it will start to freeze. Water needs to be at 0 °C to freeze.

If you warm ice up from below freezing to 0 °C, then it will remain ice, unless you add more heat to melt the ice. Ice needs to be at 0 °C to melt.

It will be the same for all substances.