[chris (OP)]

Can astronomers actually see a black hole, or even see inside a black hole, or are the presences of black holes merely inferred based on their effect on nearby space?

[jeffreyH]

Light doesn't escape the event horizon of a black hole so normal observational methods are unavailable. A black hole could be inferred by gravitational lensing of the light from objects behind the black hole. Polar jets are another phenomena that can give away the presence of a black hole.

Seeing inside a black hole is not possible since signals sent through the horizon can never return. At the horizon the escape velocity is the speed of light. This gave rise to the information paradox and the debates among physicists that followed. Ultimately this debate produced the idea of Hawking radiation and the idea that entropy related to the surface area of the event horizon. In turn this was applied to any volume in the universe where the surface of the volume related to the internal entropy. This in effect is the holographic principle.

https://en.m.wikipedia.org/wiki/Black_hole_information_paradox

[evan_au]

As Jeffrey said, you can't see a stellar-mass black hole directly (one the mass of the Sun or larger).

The nearest big black hole (about 4 million times the mass of the Sun) is in the center of our galaxy. But this region of space is hidden by thick dust clouds, making it hard to see anything.

For at least 15 years, astronomers have been observing bright stars orbiting the center of our galaxy. The speed of the orbit tells us the mass of the object they are orbiting. They don't directly tell us its size - just put a lower limit on it. This strongly suggests that it is a black hole, but it does not prove that it is a black hole, which will have a specific size (much smaller than the distance of these stars).
See: https://en.wikipedia.org/wiki/Sagittarius_A*#Central_black_hole

The Event Horizon Telescope is trying to produce radio-telescope image of the black hole at the center of our galaxy, using synchronized radio-telescopes across several continents. Microwaves can peer through clouds of dust much more easily than visible or infra-red telescopes. With more development, this should be able to measure the size of the event horizon directly, and confirm that this prediction by Einstein is correct.
See: https://en.wikipedia.org/wiki/Event_Horizon_Telescope

Recent detection of gravitational waves leaves no doubt that black holes exist, because nothing else is known which could produce the distinctive "chirp" signal of two black holes colliding. 3 collision events have been published so far. Gravitational waves are not light, but in some ways they behave similarly; I think astronomers are happy to call this "seeing" black holes, even if they can't currently point to where in the sky the event happened. (This will improve in the next 2 years, with more gravitational wave telescopes coming online.)
See: https://en.wikipedia.org/wiki/First_observation_of_gravitational_waves

There is a hypothetical possibility of micro black holes (perhaps the mass of a mountain), which could be observed directly. But nobody has been able to confirm observation of one at this time.
https://en.wikipedia.org/wiki/Micro_black_hole

[RD]

... are the presences of black holes merely inferred based on their effect on nearby space?

This is gravitational lensing of a distant blue galaxy by a closer orangy one ...
 

https://apod.nasa.gov/apod/ap111221.html

You'd need a super-massive black-hole to produce an effect like that.

[chris (OP)]

Thank you all; these insights came in very handy for informing a discussion we had on this week's programme, which included a segment on the Event Horizon Telescope. I'll post a link to the piece once it publishes (tomorrow).

[alancalverd]

The answer is in the question. If an object is "scientifically" black, it doesn't reflect or emit light, so you can't see it - you infer its existence from the absence of light.  Not a problem - we can infer all we need about a black cat, except on a dark night in a coal cellar. Fortunately the universe has plenty of photons so we can learn a lot about black holes from the bits we can't see.

[evan_au]

Before the "Weakly Interacting Massive Particle" (WIMP) theory of Dark Matter became popular, one of the leading contenders was "MAssive Compact Halo Objects" (MACHOs). This latter theory assumed that there could be many black holes, failed stars and starless planets in and around our galaxy.

Searches were conducted to look for visible stars which changed their brightness due to gravitational microlensing. This is where the gravitational field of a dark object (including black holes) caused the light from the star to be curved and magnified, as seen from Earth.

These studies concluded that MACHOs did exist in and around our galaxy, but there weren't enough of them to account for a majority of dark matter symptoms observed.

Automated surveys for gravitational microlensing events are continuing.

See: https://en.wikipedia.org/wiki/Gravitational_microlensing