This question may sound silly but I'm really not pretty sure whether stars are rotating or stationary. If they are not rotating what makes them to be stable?
Stars rotate due to the angular momentum of the gas they formed from. This angular momentum must be conserved, and remains as the rotation of the star and it's satellites. If a star collapsed from a completely static gas cloud it would not rotate, but would still be stable. The stability is provided by the hydrostatic equilibrium between radiation and thermal pressure with gravitation collapsing the star - not the stars rotation.
I don't know any statistics but I expect all stars rotate to some degree; It is usually the turbulence of gas clouds that leads to overdensities that then collapse in to stars.
Like Chris said above, stars rotate to conserve their angular momentum. They have an initial angular momentum from their formation. In a stellar system both energy and momentum, must be conserved for the system to be stable. Throughout the evolution, the system may radiate some angular momentum away in the form of gravity waves (ie. double degenerate binaries). During this process the separation of two stars will be reduced and the rotation rate of the star will spike to conserve angular momentum.