With Tom Waits, it begins with the voice. Descriptions of his vocal instrument usually invoke an ear, nose, and throat doctor’s nightmare: the sound of a man who has gargled drain cleaner or smoked cigarettes filled with home insulation. Mr. Waits has spent more than two decades borrowing from Louis Armstrong, Howlin’ Wolf, and Captain Beefheart in equal measure, but he’s put a highly personal stamp on his rough, untutored holler. He owns his sound like no other vocalist, to a degree that has been successfully tested in court (see the lawsuits he’s filed against advertisers who have used a sound-alike to sing jingles).

Mr. Waits’s voice began softer and comparatively nasal on quieter, jazz-influenced records such as his 1973 debut “Closing Time.” It gradually developed the raw expressiveness that by the 80s had become his trademark, complimenting perfectly his ramshackle junkyard orchestra aesthetic and songs of desperate people existing on the margins.

To the fans who have made Mr. Waits an icon of uncompromising avant-rock, the quirks of his singing are like the “L” train rattling outside Elwood Blues’s apartment, so omnipresent you don’t even notice them. To detractors, his voice is an obstacle that obscures fine songwriting. These folks would rather hear Rod Stewart’s cover of “Downtown Train” than the barking original from Mr. Waits’s 1985 album “Rain Dogs.” Many are in the middle, enjoying Mr. Waits as he slips into his scotchsoaked croon on gentle ballads, reaching for the stop button when he fires up the bullhorn.

For the latter crowd, Mr. Waits makes his 3-CD boxed set, “Orphans: Brawlers, Bawlers & Bastards,” easy. The majority of these 54 tracks have never been released; some are album outtakes, some were recorded only recently, others were written for singers on spec. And as the subtitle makes plain, the discs are divided by sound. “Brawlers” boasts the blues shouters and abrasive rockers, “Bawlers” contains the ballads, and “Bastards” is a collection of tribute compilation tracks, spoken word pieces, and experiments. It’s not a proper career summary, but the breadth of the music here reflects Mr. Waits’s oeuvre during the last quarter-century, highlighting its best and worst aspects.

The “Brawlers” disc is the roughest going, and not just because of its prickly sound. Since Mr. Waits moved to the indie label Anti to release “Mule Variations” in 1999, he has, like Bob Dylan, often neglected to write original melodies, preferring instead to fall back on blues and standards for structure. This is in part a byproduct of his reverence for the folk tradition and humble acceptance that there’s nothing new under the sun. But the rote structures that infected “Mule Variations” and have cropped up elsewhere since are what most clearly separate his spotty later work from his prime mid-1980s run.

Mr. Waits has admirably retained a desire to be loud and noisy even as he glides through middle age. His last album, 2004’s “Real Gone,” was perhaps his most abrasive. But listening to 16 straight songs of him in this vein on “Brawlers” can be hard to take. As the “Brawlers” disc wears on, you begin to think about what other songs by Mr. Waits these outtakes sound like, or which 12-bar progression they borrow from. So bellowing tunes here like “Lowdown” and “Fish in the Jailhouse” are best in doses of just a song or two.

The “Bawlers” set is more satisfying partly because Mr. Waits seems to work a little harder on the tunes when he doesn’t have the power of his wail to fall back on, and partly because the more relaxed musical setting allows you to absorb the nuance of his songwriting.

For a guy who started out lumped with the Lost Angeles confessional singer/songwriters, Mr. Waits’s greatest strength has always been his instinctive grasp of theater and characters. Since the 1980s, listeners have rarely heard him use the word “I” in a song and thought of Tom Waits, the man.

What “really happened” is always incidental to the movement of his musical dramas, and storytelling — coupled with his persona — is Mr. Waits’s true genius. So the best of the “Bawlers” disc, the stripped-down songs like “World Keeps Turning,” “Bend Down the Branches,” and “Jayne’s Blue Wish,” are terrifically affecting and pretty, easily the equal to his best early ballads.

Storytelling is a theme on the “Bastards” disc as well, as Mr. Waits returns frequently to the spoken word pieces that have become dependable interludes on his albums, occasionally to the point of self-parody. People have often said of distinctive vocalists (and Mr. Waits is certainly one of them) that they could read the phone book aloud and remain interesting. Mr. Waits tests that theory on “Army Ants,” on which he recites data about killer insects from an encyclopedia over a backing of plucked guitar and quickly creeping bass. That it works decently well says something about how easy these mood pieces have become for him.

“Bastards” is certainly hit-or-miss, but a few of Mr. Waits’s yarns remind us of the thing that’ll always save him, no matter where his songs go: his sense of humor. “Pontiac”is a riveting ode to automobiles he has known and loved, “Dog Treat” is a live between-song interlude about the dubious ingredients present in a canine snack, and the closing “Missing My Son” is a brilliant gag with a telegraphed ending that only someone as likeable as Mr. Waits could bring off. All hold up remarkably well to repeated listening.

There’s a lot to digest on this set, and chinks in Mr. Waits’s armor are readily apparent. He can sometimes seem like a victim of his own distinctive style, as if he has become trapped in the musical universe it has taken him two decades to build. But then his firm grounding in the fundamentals — storytelling, humor, theater — bubble up, and you’re ready for one more creaky favorite, sung rough-voiced and slightly out of tune by that goofy guy in the funny hat.