If there is such a thing as an oddball favourite opera, my oddball favourite opera may be La Gioconda by Amilcare Ponchielli. This opera, based on a Victor Hugo play with the libretto written by Arrigo Boito, has the most insane plot of any opera ever, a complete mess. It has no psychological depth whatsoever; every character is a flat-out stereotype: the noble-yet-whiny tenor, the mustache-twirling baritone villain, the jealous husband, the wife who loves the tenor but remains true to her psychotic husband, and so forth. There is really no point to the opera at all except the opportunity for six good singers to sing some good hummable tunes. Yet I like La Gioconda more than any opera by Puccini, and really more than any other post-Verdi Italian opera.
Many operas have a reputation for having insane plots, but don't really deserve the reputation. Verdi's Il Trovatore gets a lot of flak for its crazy coincidences and over-the-top melodrama, but its plot is actually quite well-structured and the characters' motivations are pretty clear; it's just that it uses a type of plot (heavy on coincidence) that was popular in 1853 -- not just in opera, but in all media -- and later fell out of fashion. But La Gioconda really deserves its reputation for incomprehensibility. It doesn't actually have, by 19th-century operatic standards, a lot of coincidences. What makes the plot impossible to follow is that the characters' motivations are impossible to understand; every character behaves in ways that cannot be explained rationally. Conrad L. Osborne recalled that when he tried to explain the plot to someone who wasn't familiar with it, the person kept interrupting him after every other plot detail to ask "But why?" And he couldn't come up with an answer.
A synopsis of the plot, such as it is, can be found at the first link, but here's a brief description of one scene, just to give you a sense of how these characters behave:
The jealous husband, Alvise, believes that his wife, Laura, has been unfaithful to him with Enzo. (Alvise thinks Enzo is a boat captain, not realizing that he's actually an exiled prince in disguise.) He confronts Laura and tells her that because she is in love with another man, she must die. He gives her a vial of poison and leaves the room, telling her that she must drink the poison and die before the gondolier outside finishes singing. After Alvise leaves the room, the street-singer Gioconda appears, having sneaked into the house and overheard everything. She anticipated that Alvise would try to make Laura kill herself, and since Gioconda has sworn to protect Laura in exchange for Laura having once saved Gioconda's mother's life, Gioconda has a plan to save Laura. She gives Laura a drug that will make her fall into a deep sleep so everyone will think she's dead, but not really.
This description covers about ten minutes' worth of an opera that lasts two-and-a-half hours. So multiply that by 15 and you'll know how truly baffling this opera is.
So what's good about it? Well, to start with, the tunes are mostly great. Unlike the Italian composers who came after him, like Puccini, Ponchielli didn't absorb much of the influence of Wagner; he didn't go for an inflated orchestral sound or try to mimic Wagner's leitmotif system the way Puccini sometimes did. Ponchielli's style is sort of a throwback to the "classic" Italian opera of the '40s and '50s, with the characters expressing themselves in simple, direct song, and with each number focusing on a single powerful emotion like love, hate or jealousy. The only other post-Verdi Italian composer whose style was so "pure," so direct, was Mascagni (Cavalleria Rusticana), a pupil of Ponchielli.
The most famous numbers include the tenor aria Cielo e Mar! L'etereo Velo, the big Gioconda/Laura catfight duet L'Amo Come il Fulgor Del Creato, Gioconda's aria Suicidio!, and of course the big ballet, the Dance of the Hours (known to everyone from Fantasia as well as Allan Sherman's "Hello Muddah, Hello Fadduh"). But probably the best part of the opera is the act 3 finale, a concerted ensemble for all the main characters: one of those ensemble numbers that allows each character's individual personality to come out even when they're mostly singing simultaneously -- and Boito even ingeniously arranges for some vital plot information to be advanced in the middle of the number.
Another great thing about La Gioconda is that whereas most operas have big opportunities for maybe three or four characters, Gioconda has big numbers, genuine star-turn moments, for six characters. The soprano, Gioconda, is definitely the star, but there are big showcase moments -- either alone on stage or with everyone raptly listening to him or her -- for the tenor, the mezzo, the baritone, the bass and the contralto (who gets to play a blind woman). The plot being what it is, there's not much time wasted in actually motivating these big star moments: as Osborne pointed out, the tenor leaves the stage at one point with very little explanation, just so the mezzo can be alone onstage and sing an aria. But when you collect six genuine stars, with big, juicy Italianate voices, La Gioconda is like one show-stopping moment after another.
Unfortunately, most performances and recordings can't gather six genuine stars, so you either have to make do with the opera as a star vehicle for the soprano (which unbalances the evening and makes it a chore to sit through some of the other characters' big moments) or a mixed bag of some singers who can stop the show and some who can't. Maria Callas was a great interpreter of the role of Gioconda, but her recordings of the opera are not particularly well-cast, so it becomes the Maria Callas Show instead of a six-star event. Among commercial recordings of the opera, the closest to a real six-star recording is probably the 1957 Decca recording with Anita Cerquetti (a star-quality soprano who burned out and retired very early), Mario Del Monaco, Cesare Siepi, Ettore Bastianini and Giulietta Simionato; only the smallest of the six parts, the contralto La Cieca [The Blind Woman] gets less than star-quality casting.