Star Trek Social Club

We probably get our best look at penal rehabilitation in Lower Decks' "A Few Badgeys More"

We learn that Daystrom Institute has a facility dedicated to evil robots, but through therapy, and exploration of art, sports, and other hobbies and psych-evaluations they may earn parole, and from there re-enter society.

Peanut Hamper made it to parole, initially as a ruse, but actually ended up taking it seriously.

Agimus is lagging behind her, but also shows signs of sincere reform.

Honestly, while a lot of it was played for laughs, I really appreciated how it really was Star Trek's optimism at its peak. People can be reformed, and are not sentenced to life in a cubical if they are capable of earning it.

From the opening log of "Whom Gods Destroy":

Captain's Log, stardate 5718.3. The Enterprise is orbiting Elba Two, a planet with a poisonous atmosphere where the Federation maintains an asylum for the few remaining incorrigible criminally insane of the galaxy. We are bringing a revolutionary new medicine to them, a medicine with which the Federation hopes to eliminate mental illness for all time. I am transporting down with Mister Spock, and we're delivering the medicine to Doctor Donald Cory, the governor of the colony.

So, at least this one TOS episode indicates that there is only one small facility which the Federation uses to house all the remaining criminally insane people in the galaxy. I think we can assume that by the galaxy, Kirk actually means the Federation. But as of that era, there apparently exists a medication that they believe will cure people of mental illness.

How much stock we want to put in one third season TOS episode I think can be debated -- and crucially we never get any confirmation as to the long term success of the medication -- but it is part of the canon.

There is also the Tantalus V penal colony from "Dagger of the Mind". Before they beam down, Kirk tells McCoy that it's more like a resort colony than a cage, though the doctor who ran the facility was using a machine to essentially brainwash both inmates and staff.

As for incarceration and rehabilitation in the 24th century, we know Tom Paris was at the New Zealand Penal Settlement when Janeway sprung him, with the approval of the Rehab Commission. When we see the settlement, the prisoners appear to be doing some sort of labour: one is carrying something, and Paris appears to be calibrating some sort of machinery. Granted, we don't know exactly what he was doing or why. Maybe he was working on a project he volunteered for or even conceived himself, and was given access to the resources to carry it out.

Ro Laren was on the Jaros II penal colony after her court martial. She was sprung from that by Admiral Kennelly, and he claims it was difficult to do so.

Kasidy Yates was incarcerated for six months for aiding the Maquis, though there's never any indication that the sentence isn't purely punitive.

In "Blaze of Glory" we saw that after his capture in "For the Uniform", Michael Eddington was being held aboard a station in a fairly small cell. He was still wearing civilian clothes. It's possible he hadn't yet been formally tried and convicted, though.

there is a depends here. Like someone else said there is the New Zealand penal colony, but it's also alluded to that a lot of criminality stems from things they have been able to treat or eliminate.

Poverty as an example is a huge driver of crime and criminal activity. Even today we have enough research to show as an area rises from poverty and people have their needs met without worry crime falls.

They apparently Identified a lot of mental disorders that lead to criminal activity and they all could be treated. The thing that scares me on this path with that future is do they basically pave the brains of the neurospicy because they think they are suffering and abnormal or do they honestly embrace it.

When you get into other species like the Klingons, there is no rehab. You repent through death mining in harsh conditions or building in places with little to no atmosphere.

The most prominent prison we've seen in the TNG era is the New Zealand Penal Settlement, but not a lot is known about it. There was definitely labour involved, though I think it's an open question whether it was "forced."