My notes from an introductory course about lasers say that
There does not exist a laser emitting in the X-ray because the spontaneous decay lifetime is too short to have stimulated emission. In fact, it goes with the inverse of the frequency of the transition, therefore being small for high frequency transitions.
I know that: $$τ_{sp} \propto \frac{1}{\omega_0^3 |μ_{12}|^2}$$ with $ω_0$ angular frequency associated with the transition and $μ_{12}$ expectation value of the transition operator. I also know that, for transition with a very low probability, such as magnetic dipole allowed (and electric dipole forbidden), this lifetime can significantly increase.
I also know that there are a lot of different selection rules (electric quadrupole, magnetic quadrupole, ...), each one less probable than the preceeding, for which the spontaneous decay lifetime could be higher.
Therefore, why don't x-ray lasers exist? Is it just that is still more convenient to develop synchrotrons or is there some other reason? What have been the scientific efforts in this direction?