What form of energy does a wind-up clock have?

Answer 1

A wind up clock contains potential energy stored in a spring.

That elastic potential energy is used to move the hands of the clock and otherwise power its operation which might include making an alarm ring. The energy in the spring primarily is used to overcome friction (producing heat) in the internal mechanism, but technically a small amount of kinetic energy is contained in the moving hands and a small amount of energy is converted to sound waves if an alarm sound is created.

To get energy into the spring, work was done on it. (Work is force times distance.) Force winds the alarm and that force is applied for a distance and in the direction of motion, even if that direction changes and it ends up being circular as in twisting a key to wind a spring. Apart from frictional forces opposing the winding, the work goes into the spring which gains potential energy. (Friction in the winding process produces some heat, so a careful accounting would put work done equals frictional heat energy plus spring potential energy. Work done equals change in the energy of the system.)

Answer 2

A wind-up clock has mechanical energy. When the clock is wound up, potential mechanical energy is stored in the clock's springs. As the springs unwind, this potential energy is converted into kinetic energy that drives the clock's movement.