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.
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.
Family Unfriendly Aesop: The storyline implies that it is justifiable to unlawfully escape from prison in a society that thinks you are a murderer, if you were wrongfully convicted. It also implies that it is justifiable to engage in all kinds of sneaky treachery, including setting up a complex scheme to launder money for one person, so long as you plan on using that same scheme to expose the person you were laundering money for as a crook. In other words, this movie implies that ends justify means. Yes, it has family-friendly aesops as well (the themes of hope and perseverance come to mind), but the moral perspective implied from some aspects of the movie (especially the scheming, underhanded nature of the protagonist) is quite at odds with more conventional forms of morality.
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).
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 Said opera music has been replaced by various kinds of music, from pop music to heavy metal, and even by flea market ads.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...