These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Bogs Diamond is the leader of a prison gang called 'the Sisters' who prey on weaker inmates to rape them again and again for as long as they like. Bogs sets his sights on Andy and together with his gang makes repeated attempts to rape and brutalize Andy over the course of several years, succeeding about half the time. When Andy tries to fight off the latest attempt, Bogs threatens to shiv Andy through the skull if he doesn't give Bogs's buddy oral sex. When Andy bluffs him out of it by playing on his ignorance, he beats Andy straight into the infirmary instead.
Warden Norton seems like a stern but affable prison bureaucrat before he reveals the true extent of his immorality and corruption. He sets up a scheme to underbid public construction projects with cheap prison labor and pockets the buy-off money from the desperate private contractors he blackmails, using Andy's financial knowledge to launder the illegal money. When a new inmate named Tommy has proof of Andy's innocence, Norton has him shot in the yard by the captain of the guard and the murder passed off as an escape attempt. He throws Andy in isolation for a month, then threatens to destroy everything Andy has built to improve Shawshank over the past 20 years in prison and make sure that he'll be handed over to the rapists again, before giving the half-mad Andy another month in isolation just to think about it.
Also, the song (Taken from The Marriage of Figaro) that Andy plays to the entire prison while inside the warden's office.
Ear Worm: The music that plays when the police are coming to get Captain Hadley and Warden Norton.
Fandom Rivalry: In 2008, a somewhat complicated series of votes on IMDb to give The Dark Knight the spot on the Top 250 as the #1 film of all time (that was from The Godfather) ultimately led to Shawshank topping the list. The Godfather had held that position for quite a long stretch of time, and its fans were, to put it mildly, not impressed; even now, over a year later, a visit to Shawshank's IMDb forums reveals thread after thread attacking it as unworthy of being #1 (and, implicitly or explicitly, favouring The Godfather).
Captain Hadley had to get some sound financial advice from Andy Dufresne in order to keep a cash windfall intact, the kind of advice that Mr. Krabs would know in his sleep. Krabs certainly wouldn't trust a banker with his money.
Memetic Mutation: The scene where Andy plays opera music for the prison has been edited many times on YouTube. Said opera music has been replaced by various kinds of music, from pop music to heavy metal, and even by flea market ads. To be fair, the reactions of the prisoners and the warden made it pretty ripe for parody.
Moral Event Horizon: The Warden ordering the death of a recently transferred con (Tommy) who has evidence of Andy's innocence, so that Andy will be forced to continue working at the prison.
Also, Hadley crossed this when he threatened to throw Andy off the roof and pretend Andy fell off.Beating up the emotionally-overwhelmed inmate earlier on could be excused by claiming he did it to deter other prisoners from making noise, but his later threat has no excuse.
Rewatch Bonus: The surprise inspection scene. The tension. We weren't aware of Andy's digging-a-tunnel project by then. When the warden talks about the poster and almost forgot to return Andy's bible...
The warden keeps his safe and all the damning evidence therein under a needlepoint that reads "His judgment cometh and that Right Soon..." (though apparently the warden's wife likes the less-preferable spelling "judgement")
Andy keeps the rock hammer that facilitates his escape hidden in the Bible, in the book of Exodus.