The maximum angular momentum, $J$, for a spinning black hole is given by
$$ J \leq \frac{GM^2}{c}\ . $$
This corresponds to a maximally spinning black hole having a "spin speed" of $\leq c$ at its event horizon, which has an equatorial radial coordinate $r \geq GM/c^2$.
Usually, we express the black hole spin in terms of a dimensionless spin parameter
$$ 0 \leq \frac{Jc}{GM^2} \leq 1\ . $$
The spin parameters of astrophysical black holes can be estimated using a variety of techniques, reviewed by Reynolds (2021). A technique that is applied to supermassive black holes at the centres of active glactic nuclei (AGN) and to stellar-sized black holes in accreting X-ray binary systems is to measure the profile of spectral lines from the inner accretion discs, that are offset and broadened by their orbital motion around the central black hole and by gravitational redshift. The amount by which the spectral line is broadened, skewed and then truncated is dependent on the spin parameter because the innermost stable circular orbit (ISCO) of material around the black hole is dependent on the spin parameter and thus determines both the orbital speed of that material and the amount by which its emission is gravitationally redshifted.
Another technique is to look in detail at the shape of the thermal continuum emission from the accretion disc. This again, shows spin-dependent differences, largely because the radial coordinate of the ISCO is spin-dependent.
A third technique that is producing results is that the detailed shape of the gravitational waveform just prior to the coalescence of binary black holes depends on the spin and spin orientation (with respect to the orbital plane) of the individual black holes.
The following picture (taken from Reynolds 2021) summarises the observational status for supermassive black holes. These appear to be a fast-spinning population. The error bars are 90% confidence limits, so there are at least some black holes though that appear to be significantly sub-maximally spinning.
There is no similar picture for stellar-size black holes, but the measurements that exist for $\sim 20$ systems suggest that they are also fast-spinning in general, although again, there are individual examples with spin parameters as low as $0.25^{+0.20}_{-0.29}$ (LMC X-3) or $0.45 \pm 0.02$ (XTE1652-453). It is unclear whether these spin rates are determined by their birth conditions or whether there is spin-up after they have formed.
The spin information from gravitational wave sources is reviewed by Tong et al. (2022). The spins of merging black holes are found to be predominantly quite low, consistent with spin parameters $<0.5$.
Are intermediate-mass and stellar-mass black holes rotating like this? Does anyone know yet?
There are no known intermediate mass black holes, so nobody knows.
Also, how much of an oblate spheroid is formed from this? Are these super-Kerr black holes substantially wider than they are tall? Can we tell from the images of M87's hole and our own (Sagittarius A*)?
The event horizon of a spinning (Kerr) black hole is more complex than the simple Schwarzschild event horizon defined by a single radial coordinate. The outer event horizon is spherical (in Boyer-Lindquist coordinates) but smaller than for a non-spinning black hole. But outside this is an oblate region called the ergosphere inside which, objects would be forced to co-rotate with the black hole.
The recently published images of the regions sourrounding the black holes in the Milky Way and M87 have the potential to be diagnostic of spin. That is because the central "black hole shadow" has a shape that in turn depends on the shape of the photon sphere around the black hole, which in turn depends on the spin parameter. For very high spin parameters, the shadow becomes non-circular and offset from the centre of the black hole (see below, taken from here - a simulation produced by Yukterez).
