One of said exceptions was the original version of Bizet’s Carmen, which actually had spoken dialogue. Later productions replaced the dialogue with recitative (a sort of speaking/singing hybrid), and now it’s a toss-up as to what you’ll see in a production. The (excellent) 1984 film version with Julia Migenes and Placido Domingo features the dialogue.
What hasn’t been mentioned is that it integrated dance into the story line. Laurie’s dream ballet expresses her feelings through dance.
There had been previous musicals where the songs advanced the story, but before Oklahoma!, they dance numbers were just spectacle. The dream ballet and some of the original dance numbers advance the plot and help define relationships.
It’s the first to integrate all the elements.
Transcribed from National Lampoon Radio Hour —
Gentleman sitting in talent agent’s office: (off-key, alternating between two notes) O-kla-homa where the wind comes sweeping down the plain, and the waving wheat can sure smell sweet when the wind comes right behind the rain —
Talent agent: Mister Hammerstein, you’re a wonderful lyricist —
Hammerstein: Oh, thank you —
Talent agent: Let me find someone to do the music, OK?
Great answers so far (and the post that @hajario linked to from an old thread adds a lot of good context).
I’ll be even bolder (if maybe not completely accurate) for the sake of clarity, and claim that prior to Oklahoma!, musicals were simply vaudeville and musical reviews, with perhaps a basic framing narrative or setting that might or might not have anything to do with what was happening on stage at a given moment.
For anyone interested - here’s a link to an episode of a podcast I occasionally listen to on just this subject! Great podcast. Fun to listen to!
As I see it, the Threepenny Opera, by Bert Brecht and Kurt Weill, from 1928, was a “completely unified and integrated piece of musical theater, in the sense that all the songs and dance numbers are directed connected to and motivated by the central storyline, and contribute to telling a single coherent story”, and the songs did not “interrupt the main story” but developed it.
Of course a musical, a musical theater play, an opera, an operette, a zarzuela and so many other words for closely related art forms are all completely different, yet in the end, the same. I would claim that Oklahoma! was groundbreaking in that it was a bigger commercial success, aided by the new-ish medium colour pictures with sound. And it was a big commercial success because the songs and the libretto were fine, and it was the right time for it. Of course I found it boring as a child, but to be frank it was not made for children who don’t understand English. I wonder now why Spanish cinema theaters showed those films (and 7 brothers for 7 sisters and the blizzard of Oz) at all. Well they did, and left a confused me back.
Oklahoma! didn’t make it to the big screen until 1955. By then it had already had a huge influence on stage and screen over the previous 10+ years.
The Wizard of Oz, I assume. The Blizzard of Ozz is an Ozzy Osbourne album. 
Mr. Steyn remarks on the deceptive simplicity of Richard Rodgers’ unforgettable tune: “It seems as simple as a folk song. But no anonymous farm-hand’s sing-a-long would have [the protagonist] ‘Curly’ sing ‘morning’ on D-natural, and then, in the equivalent spot two lines on, sing ‘feeling’ on D-sharp. You need a professional for that: they signal Curly’s intensity of feeling about the land he belongs to . . .
In other words, Oklahoma’s score brought tactics from opera and jazz to a Broadway that previously was focused entirely on mainstream popular songwriting techniques.
According to one theater history article I read years ago, the opinion among the Broadway wags when Oklahoma was in rehearsal was, “No girls, no gags, no chance!” After it opened, that changed to, “No girls, no gags, no tickets.”
You guys might enjoy this video comparing Oklahoma to its source, Lynn Riggs’ play Green Grow The Lilacs. It’s by Christi Esterle, aka Diva of Musical Hell, a video series that reviewed bad movie musicals. Christi no longer does Musical Hell, but she still does her other series, like Know The Score which discusses musicals and their tropes, and At The Source, which compares musicals to their source material.
Which became a whole subgenre of Broadway songs. (“Whatever Lola Wants,” “No Time at All,” “Big Spender”…)
Those examples actually indicate how comparatively rare the theme is: “Whatever Lola Wants” is sung by a character deliberately trying to lure the male protagonist in accordance with orders from her boss; “Big Spender” is likewise a pitch by entertainers/sex workers for whom luring men is part of their job. “No Time at All”, sung by a 66-year-old character, is the only one of those three that’s plausibly a sincere expression of enjoyment of sex.
I think light opera is a term that can include works ranging from an American musical (at least that requires pretty good voices, not all do) to a Mozart singspiel (like Abduction from the Seraglio). That’s my opinion, of course, it’s really a catch-all term designed to appeal to non-opera lovers, assuring them that they won’t fall asleep during it.
to me the definitive Carmen!+++
In the First Act, when Curly went to visit Jud, did Jud confess to murdering a bunch of people? He speaks of leaving town after a farmhand, spurned by his love interest, burned down a barn/home with the people inside. Was Jud confessing to murder? Just telling Curly that he’s capable of murder? Just making polite conversation?
Here’s the relevant part of the script:
JUD. If it ever come to gittin’ even with anybody, I’d know
how to do it.
CURLY. (Looking down at gun and pointing.) That?..,
JUD. Nanh! They’s safer ways then that, if you use yer
brains… 'Member that far on the Bartlett farm over by
Sweetwater?
CURLY. Shore do. 'Bout five years ago. Turrble accident.
Burned up the father, and mother and daughter.
JUD. That warn’t no accident. A feller told me - the h’ard
hand was stuck on the Bartlett girl, and he found her in
the hayloft with another feller.
CURLY. And it was him that burned the place?
JUD. (Nodding.) It tuck him weeks to git all the kerosene -
buying it at different times - feller who told me made
out it happened in Missouri, but I knowed all the time
it was the Bartlett farm. Whut a liar he was!
CURLY. A kind of a murderer, too. Wasn’t he?