A lot of fans of the 1951 film like to play up the notion of Blanche being a variation of Scarlett O'Hara from Gone with the Wind; this is mainly because Vivien Leigh played both roles, as well as the urban legend that Leigh got the part SPECIFICALLY because Elia Kazan wanted to play around with the idea of "What if Scarlett O'Hara Lost Her Mind" as his take on Blanche. Directors of the original play are rather divided on who's right, and it shows in the performances.
Blanche is obviously the protagonist, but seen from one point of view, she does come in to Stella and Stanley's "happily ever after" and mess everything up. People tend to forget that she is, in fact, a guest and therefore, in some instances at least, completely in the wrong.
You could argue that Stanley is almost as insecure as Blanche, deep down inside. He seems sure that Blanche could potentially tear Stella away from him, and when he realizes what he did in a fit of rage, he breaks down in tears and calls her back, quoting from the stage directions, "with heaven-splitting violence". His tough persona is merely a facade to hide his deep insecurities and fears.
A rather interesting point with this is that nowhere in the script does it say that Stanley raped Blanche. It only says that she faints (meaning she wouldn't know what, if anything, happened anyway) and he "carries her to the bed". This means that this scene (and therefore Stanley's character) could be interpreted one of two ways: either he did rape her while she was unconscious or he only meant to frighten her, and didn't go any further than that.
It's up in the air how we're meant to see Stanley's Relationship Sabotage between Mitch and Blanche. On the one hand, it's a dick move - and Stanley clearly told an embellished version of the events. On the other, Blanche was lying to Mitch and Stanley didn't want his friend to be saddled with someone who would definitely be a huge burden.
Stella is seen as a victim of domestic violence, but one scene in particular where Stella describes Stanley smashing all the lightbulbs on their wedding night suggests she might be attracted to Stanley's violent qualities. If you listen to how she talks, she almost sounds as if she's turned on. Note that she never seems to have a problem standing up to Stanley and she's not afraid to shout back at him. It's entirely possible Stella could be addicted to the drama that Stanley brings, and is just as messed up as him in her own way.
Take the scene after the poker game - the "Hey, Stella!" sequence. Stella goes back to Stanley after he hit her. Is this Stockholm Syndrome on Stella's part? Or is Stella the one with the power here? In the film at least, Stella looks like she could be enjoying making Stanley beg for her to come back - thus making this a sort of Masochism Tango.
Award Snub: Despite being lauded as one of the most powerful performances of all time, Marlon Brando's performance as Stanley was not awarded an Oscar (and the film did sweep the other three acting categories, with Vivien Leigh, Karl Malden, and Kim Hunter all winning in their categories.)
Draco in Leather Pants: The role of Stanley, a truculent, wife-beating rapist, is what propelled Marlon Brando into sex symbol status. He does have sympathetic qualities, mind you, but plenty of bad ones to equal them.
Fridge Brilliance: Why did Stanley serve in the Engineer Corps during World War II instead of a regular combat unit? Well, a common stereotype of Poles is that they're good at mending things, and one of the roles of Army Engineers is to design and construct dams and flood prevention systems to prevent flash floods.
Fridge Horror: Considering that this play was written in the 1940s, it's a very real possibility that Blanche would have been lobotomized after the story's end...
Heartwarming in Hindsight: Stanley and Blanche's cordial first meeting - since Marlon Brando and Vivien Leigh reportedly became quite good friends during filming.
Jerkass Woobie: Blanche. She's very harsh towards Stanley (who she sees as a 'Polack' and an 'ape'), and she isn't particularly nice in general, but once you find out about her past, it's very hard not to feel sorry for her...
Moral Event Horizon: Stanley's MEH is generally considered to be raping Blanche, but you could also argue that he crosses it when he tells Mitch about Blanche's exploits (thereby ruining their relationship), or when he presents Blanche with his 'birthday present'... a bus ticket back to Laurel.
Some extra Fridge Horror here: during the Poker night, Stanley loses self-control and ends up hitting Stella. Considering how quickly she forgives him and how calm she is about it, it seems that it's a common occurrence, and studies show that parents who are domestic abusers will have kids that will grow up to be domestic abusers. If you believe the Alternate Character Interpretation above that Stella could enjoy making Stanley beg for her to come back, it makes their relationship even creepier.
The scene leading up to Blanche's rape at the hands of Stanley and the actual implied rape.
Blanche herself, who is seen as an Asshole Victim nowadays for manipulating her sister, constantly living in denial even in the face of the truth, an absolute inability to do anything useful, mooching off both Stanley and Mitch, being an ephebophile and getting kicked out of her position as a teacher for it, telling lies to everyone she comes across (and especially her long line of lovers). There was a case where, during a modern revival of the play, people cheered as Blanche was raped.
On the other hand, Blanche is clearly mentally ill and needs legitimate professional help, yet she has the misfortune to live in a time where the only "help" offered to people like her is lobotomy, and her only family in the world is in an abusive marriage with a man who wants to break her into even smaller pieces than she's already in, and send those pieces packing. (And her sister sides with her husband after he rapes her, and sends her to said lobotomy rather than leave him.)
Aside from that, most characters treat her promiscuity as just as bad as, if not worse than, her mental instability. In today post-sexual revolution society, women are not quite as morally condemned for having lots of partners as in her time. (Although sleeping with an underage student is seen as worse today than back then.)
Values Resonance: One of the many layers of the play is Blanche and Stanley are pretty symbolic of an "Old South vs. New South" dynamic. Blanche was a faded relic of the bygone Antebellum South (she is, after all, played by Scarlett O'Hara herself,) while Stanley was a modern, industrial, blue-collar immigrant, symbolic of where the South was going. An important discussion when the play was written in 1947, since the South was still rebuilding after the war. However, that same "Old Guard vs. New Blood" debate is showing up all over again, with the newest generation of Southerners being much more diverse and progressive, and frequently at odds with the more conservative previous generation, believing that clinging to the Good Old Ways just isn't working anymore.