I'm aware that it was illegal as it went against the ideas of their "Socialist Morality", but how widespread and available was it? How was it acquired? Was it similar to mid-20th Century America in its underground nature?
Oh my god! A porn question!
But sadly, not one I have much expertise on--I could tell you some about pre-Soviet pornography, but USSR is way too late for my knowledge. However, no fear, I know where you can start.
If you can get your hands on International Exposure the essay "Stripping the Nation Bare: Russian Pornography and the Insistence on Meaning" focuses exactly on that. It's not particularly my field, but I know this is a well-respected essay on the topic. And Borenstein's sources will give you even more to go on.
How it happened that I became an expert in Soviet pornography now? :-)
it went against the ideas of their "Socialist Morality"
There were heated pro- and contra- debates several times and every time contra- side won:
right after October revolution, a sizable number of revolutionary intellectuals supported ideas of "free sex", "abolition of traditional family", "cult of naked body" etc. It led to severe backlash among conservative peasantry, workers and was exploited to great effect by counter-revolutionary propaganda (both internally and abroad), so Soviet government moved quickly to shut down any signs of endorsement of such policies from Bolsheviks.
During NEP pornography and prostitution weren't prosecuted and de-facto came close to being legal. Only underage prostitution/pornography was effectively prosecuted as "children exploitation". Note that in 20s there were a lot of street kids, orphaned by Civil War, hunger and diseases like typhoid fever, dysentery, influenza, cholera.
In 30s there was attempt at using pornography to increase birthrate among city dwellers (birthrate has quite high in countryside)
In 60s-80s some movies have quite frivolous bed scenes. Female nipples, asses of both sexes were shown. Quick frontal nudity happened too. I'd guess these movies would be censored in US nowadays. Some heads rolled and some movies were verboten but not all (heads/movies).
Late 80s - till USSR dissolution. Despite porno law being still in effect, you could see all kind of nakedness in every kind of media. Theater, movies, newspapers (official CPSU publications included), magazines. To a point that it generated tiredness and backlash from populace by mid-90s (but that was in post-USSR world).
How was it acquired?
Mostly from street peddlers. Two major classes were B/W domestic production and smuggled in western adult magazines. Domestic production was of low quality (manually developed photos, cut and bundled sometimes, mostly pocket-sized - playing cards, calendars etc.). Peddlers were often from deaf|mute cooperatives (USSR-specific form of organizing disabled citizens with similar disabilities) and they usually sell on trains - harder to catch and impossible to interrogate about source. Selling or just giving away to acquaintances was also widespread.
Was it similar to mid-20th Century America in its underground nature?
Since I have no idea how porno was treated in mid-20th Century America - sorry, can't say.
In USSR, YMMV - you could get a steep fine and|or jail sentence or get away with citation|reprimand letter to the job. Depends a lot on personalities and seriousness of the offense. One thing for a youngster to be spotted with a porno-calendar and another to be caught with fresh batch of such production at a place of production. Worst would be caught with western porno magazines and god forbid if any of them have anything about USSR - that could get you much more serious article of law ("anti-Soviet propaganda" most likely) with pornography being aggravating circumstance. 10+ years of hard work.
Sources
NSFW - Soviet B/W amauter porno, 70s-80s Be careful as some ads on this site might contain malware
NSFW - Merkurov. Erotic Alphabet, 1931 Note that Merkulov later became acclaimed artist, was member Union of Artist (Союз Художников СССР) - most prestigious club of artists in USSR complete with fellowship, housing and great vacation spots. He never draw anything like that though.
NSFW - Erotic scenes in soviet movies from 30s in B/W to late 80s in color. Many movies were very popular and had multiple re-runs.
NSFW - Another collection of erotic scenes in soviet and post-soviet movies Some overlap with [3].
Have a look at this old thread. While there aren't too many good answers, it's got a bit of info.