A main idea for observations of astrophysical phenomena related to the (canonical) Penrose process regards the "collisional Penrose process." The canonical Penrose process involves the decay of a single particle for extracting energy from the ergosphere, but this is known to not be particularly efficient. "Collisions" of particles in the ergosphere are potentially more efficient, as "the 'collision' of multiple particles can reach arbitrarily high center-of-mass energy in the limit of extremal black hole spin." One possibility for astrophysical observations is the "potential to enhance annihilation of dark matter particles in the vicinity of a supermassive black hole."
From the discussion section of the paper by Schnittman:
Despite the wide variety of fundamental and fascinating results described in the
previous sections, by most accounts the collisional Penrose process is unlikely
to play a significant role in astrophysical processes. Even the highest efficiency
reactions can only provide energy boosts on the order of a factor of ten or so, far
below the ultrarelativistic particles seen from gamma-ray bursts or active galactic
nuclei1. And in any case, even those moderately high-efficiency events require such
fine tuning of initial conditions, they are probably impossible to realize in a natural setting. One potential (although admittedly speculative) exception is the annihilation of dark matter (DM) particles in the ergosphere around a Kerr black hole.... The simplest annihilation model produces two photons
of equal energy and isotropic in angle in the center-of-mass frame of the reacting
DM particles. These photons are then propogated to an observer at infinity, where
they can be summed to produce images and spectra..... If we take the threshold center-of-mass energy to be moderate, we find that most of the annihilation photons are produced within the ergosphere region, and are thus sensitive probes of the Penrose process.
A black hole that is embedded in a uniform magnetic field can give rise to an "electromagnetic Penrose process" but it is unclear whether these effects would be observable, for example in the polarization of light as observed by the Event Horizon Telescope.
The second idea is regarding the origin of relativistic astrophysical jets, but this does not directly involve the canonical Penrose process. The currently accepted explanation for jets is the Blandford–Znajek process, although there was quite a bit of controversy surrounding it in the 1990s. The main alternatives to this process for explaining jets all have in common the so called "magnetohydrodynamical Penrose process" which is not a Penrose process per say but is instead inspired by it. From this paper:
In the original Penrose process such particles are created
via close-range interaction (collisions, decay) with other particles,
which gain positive energy and carry it away [see Williams (2004)
for modern attempts to develop this idea in applications to astro-
physical jets]. However, electrically charged particles can also be
pushed on to orbits with negative energy by the Lorentz force, and
this is what makes possible the so-called ‘MHD Penrose process’.
In the MHD Penrose process, the ergospheric plasma and the magnetic field play similar roles to those of the negative and positive energy particles in the canonical Penrose process. See the discussion section of this for details of why the canonical Penrose process is considered to be a special case for single particles such as photons, high frequency electromagnetic waves, and dust. The ergosphere is crucial for the Blandford–Znajek process, but it is an uncertain matter theoretically. In the future, investigations of jets may shed light on the canonical Penrose process, or other similar processes.