Do astronomers have evidence for observation of a stationary black hole?
The simplest kind of black hole is a Schwarzschild black hole, which is a black hole with mass, but with no electric charge, and no spin. So Schwarzschild black holes are stationary black holes.
Black holes are not expected to be charged, since the material they are made of and the material they accrete are almost certainly electrically neutral to an extremely good approximation.
However, they are expected to spin, both as a consequence of conservation of angular momentum during their formation or from subsequently accreting material with specific angular momentum.
I am aware of at least two methods for estimating black hole spin: from gravitational wave observations of merging black holes and from the X-ray spectra of supermassive black holes at the centres of active galaxies and quasars.
The spins of merging black holes leave a second order effect on the gravitational wave signature (the dominant effect comes from the masses of the black holes). The LIGO detections of black hole mergers were able to put weak constraints on the initial spins of the two black holes In Abbott et al. 2015, the initial black hole spins for GW150914 are $0.32^{+0.49}_{-0.29}$ and $0.44^{+0.50}_{-0.40}$, where the error bars indicate 90% confidence regions and the spin is expressed in terms of the dimensionless spin parameter that can vary from 0 (Schwarzschild black hole) to 1 for a maximally rotating Kerr black hole. The second LIGO detection was more poorly constrained; Abbott et al. 2016 say that one of the initial black holes must have a spin $>0.2$. In both cases, the final merged black hole had a very significant spin.
Hot, completely ionised iron in the gas close to the centres of accreting supermassive black holes gives rise to emission lines that are shaped by the metric of space near the black hole, producing asymmetries in the line profile. Fitting these is a difficult task and somewhat model dependent. A set of 11 objects that were studied by Patrick et al. (2012) yielded a few objects with spin parameters of $\sim 0.7$, but also a few where only upper limits could be found - the lowest was $<0.3$.
So at the moment there is no strong evidence for Schwarzschild black holes in nature and they are unlikely to exist since it is most probable that black hole have formed with, or have accreted, some angular momentum.
