The theme from High Noon by Dimitri Tiomkin and Ned Washington lays out the hero's motivation and objectives while appealing to his wife to stay with him in spite of what he's about to face. Despite the line "look at that big hand move along nearin' high noon," it's more expository than a Title Theme Tune. Both the High Noon theme and 3:10 to Yuma (the 1957 version) were sung by Frankie Laine (who also sang the title-song for "Gun Fight at OK Corral").
And since High Noon was a major source of parody for Blazing Saddles, Mel Brooks got Frankie Laine to sing his theme song too!
Blazing Saddles not only has the aforementioned tune at the opening credits, but the first scene with the townsfolk of Rock Ridge begins with a song describing how peaceful the town was until the bandits invaded. It ends with this verse (revealing that the song was sung in a church by the congregation):
Now is a time of great decision, Are we to stay or up and quit? There's no avoiding this conclusion, our town is turning into shit.
The movie High Society features an opening song explaining the premise sung by Louis Armstrong.
Night Watch: The original Russian movie adaptation had an expository theme tune during the credits, summarizing the entire movie in a humorous pseudo-rap. The international version omits it.
The song at the beginning of Pirates of the Caribbean: At World's End gives the back story in a symbolic form. This provides replay value: you already have to know things revealed later in the movie to have a chance of getting it.
One for each other and all for one the Three brave Amigos are we Brother to Brother and everyone A brave amigo wherever they need us, our destinies lead us amigos we're always together wherever we go we're three brave amigos and we'll be amigos forever we are the Three Amigos we are the Three A-migos we are the Three Aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa migos! And Amigos forever we'll be!
"dum dum dum dum dum dum dum dum FLASH! — AAAAA! Saviour of the universe! dum dum dum dum dum dum dum dum FLASH! — AAAAA! He's a miracle! dum dum dum dum dum dum FLASH! — AAAA! King of the impossible! He'll save every one of us Stand for every one of us He'll save with a mighty hand Every man, every woman, every child With a mighty flash!
Spaceballs had a catchy one (doubling as a Villain Song), but being a comedy, it was placed near the end of the movie (too late to do any real exposition), and as the titular villains were scrambling to evacuate their ship.
If you're living in a bubble and you haven't got a care Well, you gonna be in trouble, 'cause we're gonna steal your air! 'Cause what you got is what we need And all we do is dirty deeds We're the Spaceballs! Watch out, we're the Spaceballs! We're the masters of space Hey, don't mess around with the Spaceballs
Some folk we never forget Some kind we never forgive Haven't seen the back of us yet We'll fight as long as we live All eyes on the hidden door To the Lonely Mountain borne We'll ride in the gathering storm Until we get our long-forgotten gold
Far over the misty mountains cold To dungeons deep and caverns old We must away ere break of day To seek the pale enchanted gold. The dwarves of yore made mightly spells, While hammers fell like ringing bells In places deep, where dark things sleep, In hollow halls beneath the fells. For ancient king and elvish lord There many a gleaming golden hoard They shaped and wrought, and light they caught To hide in gems on hilt of sword. On silver necklaces they strung The flowering stars, on crowns they hung The dragon-fire, in twisted wire They meshed the light of moon and sun. Far over the misty mountains cold To dungeons deep and caverns old We must away, ere break of day, To claim our long-forgotten gold. Goblets they carved there for themselves And harps of gold; where no man delves There lay they long, and many a song Was sung unheard by men or elves. The pines were roaring on the height, The winds were moaning in the night, The fire was red, it flaming spread; The trees like torches blazed with light. The bells were ringing in the dale And men looked up with faces pale; The dragon's ire more fierce than fire Laid low their towers and houses frail. The mountain smoked beneath the moon; The dwarves, they heard the tramp of doom. They fled their hall to dying fall Beneath his feet, beneath the moon. Far over the misty mountains grim To dungeons deep and caverns dim We must away, ere break of day, To win our harps and gold from him! Far over the misty mountains cold To dungeons deep and caverns old We must away, ere break of day, To find our long-forgotten gold.\\
"The Bells of Notre Dame" from the Disney adaptation of The Hunchback of Notre Dame, which is sung in the context of Clopin telling the story as a puppet show for the kids of Paris. It is not only reprised in the ending, but also introduces several leitmotifs for the film's soundtrack, most notably the wordless choral Notre Dame motif that becomes the basis for "Heaven's Light/Hellfire" (the latter giving it the With Lyrics treatment).
"Arthur's Theme (Best That You Can Do)"'s first verse and chorus are about someone suddenly, quickly falling in love and coming to the realization that it's the best thing that's ever happened to them, which is the title character's situation. The second verse is specifically about that character, a Lonely Rich KidMan Child:
Arthur, he does as he pleases All of his life, his master's toys And deep in his heart he's just, he's just a boy Living his life one day at a time He's showing himself a really good time He's laughing about the way they want him to be