Question

When is the velocity of a mass on a spring at its maximum value?

Answer 1

It depends whether or not the mass is attached. However there are three states of a spring (compressed, equilibrium, extended). Both the compressed and extended states tend to bring the mass back to the equilibrium state, i.e. they exert a force to do so. The velocity is greatest before the compressed state, and (after compression) before the extended state. The transition from compressed to equilibrium is the only state which will increase the velocity of the object.

Answer 2

I should add that extended -> equilibrium also increases the velocity, if the object is attached.