X-ray Lasers and Forbidden Transitions 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? 
 A: As mentioned in Semoi's answer, electronic transitions in the x-ray regime have the disadvantage that they will tend to be ionizing transitions, i.e. they will put one electron in the continuum, where it will tend to fly away and not come back.
However, in general, that is only true for neutral atoms, but once you remove one or a few electrons, the remaining electrons are much more tightly bound, which means that you can have transitions with a much higher energy difference that still remain within the bound-states manifold. Thus, if you're working in an ionized plasma, you have good chances of being able to implement a closed lasing cycle, which can be pumped externally or even via the plasma's own collisional excitations.
Of course, this will make for a challenging experiment. For one, free electrons can be highly dispersive in the soft x-ray regime, so phase-matching needs to be done carefully. More importantly, good optical elements, particularly in transmission and at normal-incidence reflection, are thin on the ground from the XUV upwards, so building a resonant cavity will be somewhere between hard and impossible (though as I explained in this related answer, the loss of the cavity is not completely fatal). Nevertheless, it can be done.
I first became aware that this sort of amplification is a possibility via the absolutely heroic experiment described in

Demonstration of circularly polarized plasma-based soft-x-ray laser. A. Depresseux et al. Phys. Rev. Lett. 115, 083901 (2015).

but a better place to learn more is their key reference,

Multimillijoule, highly coherent x-ray laser at 21 nm operating in deep saturation through double-pass amplification. B. Rus et al. Phys. Rev. A 66, 063806 (2002).

which itself reviews many viable approaches to the problem.
A: Consider an electron, which is bound to an atom. If it absorbs an X-ray photon, the electron leaves the atom and becomes a free electron. Hence, the inverse process is "not possible" by stimulated emission -- the free electron does not belong to the atom. Furthermore, the X-rays would "immediately" be re-absorbed, because transitions of bounded electrons into the continuum are not constrained by transition rules. 
A: The X-ray laser is more of a stimulated Brehmsstralung emission or synchrotron emission process. If you accelerate lasers with a frequency $\nu_a$ then the radiation emitted is approximately $\nu~\simeq~\gamma^2\nu_a$. The wiggler or free electron laser operates on this principle. An array of magnetic dipoles force a beam of electrons to wiggle or follow an undulating path. If the velocity of the electron is $v~-~0.999999c$, or $\gamma^2~=~5\times 10^5$. Now assume the dipole magnets are $1\:\mathrm{cm}$ apart. The frequency of oscillation for the electron beam will be about $\nu~=~3\times 10^{10}\:\mathrm{s}^{-1}$. The frequency of radiation emitted is then about $1.5\times 10^{15}\:\mathrm{Hz}$. This is in the low UV range. This requires the electrons be pushed to $\gamma~=~224$ or to $112 \:\mathrm{MeV}$.
