If a ball stays on the ground, it has no height and no velocity. So what energy does it have kinetic or potential?
As you've worked out, since $v=0$ then the kinetic energy must also be zero.
Potential energy is a little more dubious. At school you are usually taught that the gravitational potential energy is $E=mgh$ but that's not quite right; this equation is the work that you must do to lift an object of mass $m$ a height $h$ or, equivalently, the work done by gravity in accelerating that object downwards by a height $h$.
An object on the surface of the earth still has gravitational potential energy. The most intuitive way to recognise this is as follows: if a sinkhole spontaneously formed beneath the ball then what would happen to the ball? It would accelerate down into the hole. But where did that kinetic energy come from? It came from the gravitational potential energy stored in the ball.
Firstly, when a body has no potential energy with respect to a potential present, in your case is the gravitational potential, then you are actually putting that object at the reference position. Next, if it has zero velocity then essentially it has no kinetic energy. Now you are left with another energy that can be measured by taking into account relativistic effects and is present all the time as an inherent part of the mass. This is the rest energy, given by energy-momentum relation:
Here $m$ is the rest mass of the body. So when velocity is zero so is momentum by the formula $p=mv$. Hence the above energy-momentum relation reduces to $E=mc^2$ which is the energy the body would posses at rest. Hope this clarifies.