When Rickie Lee Jones emerged full-blown onto the pop landscape in 1979, it hardly seemed possible. Her self-titled debut contained a hit song, “Chuck E’s in Love”, and a half dozen other instant classics that bore the signature of a fully-formed artistic vision. Jones combined the jazzy waifishness of Joni Mitchell with the streetwise grit of Tom Waits—and her act was a triple threat of brilliant singing, top songwriting, and ingenious arrangement.

Jones’ only flaw was her timing. Punk had just swept through, clearing away not only the overblown corporate schlock-rock but also much of the public’s taste for smartly arranged, harmonically sophisticated pop. Keen hitmakers like Steely Dan took the following decade off while plotting a comeback. But Jones was not deterred. Her sophomore disc, Pirates, was a masterpiece of complex arrangement and heartfelt singing, an album of great storytelling filtered through Steely instrumentation. But it seemed as if no one heard it. The ‘80s—which maybe should have been Jones’s decade—belonged to The Material Girl and the likes of Flock of Seagulls.

But maybe that was OK. Despite “Chuck E”, Jones’s trajectory was never about selling more records, even if her canvas was the classic American pop recording. Because of her voice (always elastic, sometimes strange, deeply intimate) and her songs (detailed narratives of dislocation, trouble, and unusual intimacy), it soon became apparent that she was making a very refined and carefully produced body of art songs. Not punk, not “new wave”, and certainly not dance-pop, Jones just plowed ahead making great music.

The Duchess of Coolsville is a three-disc appreciation of a brilliant career in music. Discs one and two are an anthology of tracks from her studio albums of mainly original songs. The tracks are arranged—perversely, I suppose—in alphabetical order, as if Jones is saying, “Hey, man—the order of the songs is pretty much irrelevant.” The third disc is a collection of demos, live tunes, and guest appearance tracks—not really A-list stuff but material that illuminates Ms. Jones’s talent in remarkable ways.

The third disc starts with a version of “Sunshine Superman” from the Party of Five soundtrack. Of course an artist like Rickie Lee Jones is recording older hipster’s like Donovan, seeming particularly comfortable with this modified blues tune. But the next song shows how her range reaches even further back—a duet on the jazz standard “Makin’ Whoopee” with Dr. John. Rickie Lee is still Rickie Lee, but suddenly her voice rings like Betty Carter to Dr. John’s variant on Brother Ray. Jones scoops and slides but always reaches her note, easily standing up to the big band accompaniment. This is followed by two contemporary jazz performances. “Autumn Leaves”, a duet with bass player Rob Wasserman, and “Atlas’ Marker”, a live track with jazz guitarist Bill Frisell’s trio on an original, both match Jones with a jazz minimalist. These tracks (along with the live “My Funny Valentine”) make the case that Jones’s base-line talent is really that of a jazz singer—someone who embellishes melody with a sense of swing and blues and who approaches the sonority of her voice with distinctive flexibility.

On the first two discs of material, that jazz and hipster sensibility is strongly represented. The obvious selections—like “Bye Bye Blackboard” featuring brilliant tenor sax from Joe Henderson—sound like jazz, but so do the Waits-y tracks like “Weasel and the White Boys Cool”, the impressionistic pieces like “Coolsville” and the more mature tunes like “Bitchenostrophy” and her cover of “Cycles”. Heard in this light, the fingersnaps and syncopated punch of “Chuck E’s in Love”—with its scat licks, hip tempo changes, Ernie Watts/Tom Scott sax part, and Steve Gadd drumming—is easily as “jazz” as anything else produced in the ‘70s.

The bonus material also makes a case for Rickie Lee Jones as the consummate singer-songwriter. The demos for “Easy Money”, “Rodeo Girl” and “Satellites” (among others) take you inside Jones’s thinking as a songwriter. “Rodeo Girl” is so complete and lovely in the demo—fully orchestrated by Jones’s overdubbed vocals and enmeshed in an atmosphere of guitar and percussion—that it’s a wonder that her producer bothered to rerecord it. Thus, you return to the first two discs with a sense of how complete Ms. Jones’s songs are before they’re given the pop music treatment. “It Must Be Love” (from the underappreciated The Magazine), and “A Tree on Allenford” (from the relatively recent The Evening of My Best Day) are great songs that other musicians would record if they weren’t intimidated by Jones’s voice and distinctive style. Everything from Pirates (six untouchable tracks) is classic, and you only wish there were more (particularly “A Lucky Guy”). Listen to “Chuck E” in this light and the hit song isn’t so much jazz as Paul Simon-level pop, the kind of song you’ll never tire of.

More likely, though, you’ll listen to all of The Duchess of Coolsville and come away shaking your head with admiration for Rickie Lee Jones’s stubborn eccentricity. With both pop stardom and jazz seriousness beckoning, you sense that she never considered any path than her own. And you also sense that this is a distinctive American musician who hasn’t quite ever been completely heard, at least not by younger people who couldn’t help but think that punk’s filtering out of so much musical bullshit was the only authenticity detector they really needed. You want to ask “classic rock” radio to stop playing “Chuck E”—which has essentially nothing in common with “Stairway to Heaven”—and replace it with that British guy married to Gwyneth Paltrow.

In era where authenticity and sensitivity in pop music is well described by Coldplay, you just wish the real things—like Rickie Lee Jones—were more widely available. This collection lays it all out for those inclined to listen.