I can't remember the name of the film I'd nominate for this dubious honour (and maybe it's Too Many Husbands - I'm pretty sure it was in one of those Icons of Screwball Comedy sets), but it's the ne plus ultra of the really fucking annoying screenwriting crutch of characters not simply explaining basic situations to their co-characters but preferring to get tangled up in ridiculously convoluted and risky charades that could be sorted out at any time by simply telling the truth, admitting an innocent mistake, etc. In the film I'm thinking of, one of the characters would evidently rather be convicted of murder and sent to the electric chair than give somebody the satisfaction of saying she loves him (or somesuch tomfoolery - I just wanted to see the idiot fry).

It doesn't quite match up to your description but True Confession has Carole Lombard pleading guilty to murder first so she wouldn't have to admit to her husband she was working outside the home and then to bring her husband glory as a defense attorney

I blame my exasperation, but I'm not surprised to discover that the big cover-up was even more hopelessly banal than I remembered! And that's in the Lombard box set, isn't it?

As ridiculous as True Confession is, I couldn't help but enjoy the manic energy of the cast, especially Lombard. Although my blind love of anything Lombard is probably coloring my opinion more than anything.

I watched it recently and it wasn't until Barrymore showed up that I realized I'd actually already seen it years ago. Lombard is often lovely, but like most contract stars she was in her share of dogs, and this pic ranks a place in the kennel

Laughter (Harry d'Abbadie d'Arrast, 1930)

I don't think this film is particularly rare - for starters, both MoMA and UCLA list 35mm prints in their catalogs, and I know the latter has loaned theirs out to at least one recent screening on the West Coast - but this film remains unavailable on video (it was never released on DVD, much less Blu-Ray or VOD).

A crappy (and IMHO unwatchable) version is on YouTube, but I finally found an AVI rip of what appears to be a tape dub of an old broadcast, complete with pauses during what appears to be cheap commercial break bumpers, which simply show the film's title. Not a great looking copy, but still watchable.

Frederic March's character alone is a prototypical element of screwball comedies, but the film as a whole doesn't quite crystalize into one, partly because it never shakes off its melancholy and sobering subtext, probably by design. It's a film thoroughly disillusioned by the emptiness of material wealth, and it's probably the philosophical precursor to George Cukor's 1938 film Holiday, which may have been based on another 1930 film but was rewritten by one of Laughter's screenwriters, Donald Ogden Stewart. Much of the film goes by with little, if any, attempt at humor, and even when it's there, it's usually brought down by a measure of sadness.

Production design is classy and often wonderful to look at, and except for a few spots, the limitations of early sound films seem to have been overcome. (There's even a striking echo that occurs in one long shot when Frederic March walks into the background…perhaps the result of necessity than planned, it does remind one of similar moments in Citizen Kane.)

It still feels a bit more dated than other Hollywood masterworks from the same era - for starters, Lubitsch's best films from around the same time have probably aged much better. I hesitate to call Laughter a great film, but it's very, very good and certainly undeserving of its obscurity.

the Doctor Takes a Wife (Alexander Hall 1940) The most laugh-out-loud funny film of this slate of flicks, ambitious professor Ray Milland finds himself fake married to Loretta Young's feminist author via a series of misunderstandings. Young must then explain to the readers of her best-selling book on the virtues of spinsterdom why she seemingly turned her back on her message by getting hitched-- by embracing the marital course, of course! A series of clever set pieces and wonderful chemistry between the leads make this and the Feminine Touch the easy MVPs of this round. Highly recommended.

the Feminine Touch (WS Van Dyke 1941) One of the most pleasant surprises of the checklist approach to these screwballs. Fast-paced and delightfully naughty, this mismatched lovers piece features a career-best comic perf from Van Heflin as a lecherous publisher who is so sex-crazed that Kay Francis literally has to lock him in a room at one point to stop him from trying to bed every woman in sight. Madcap farce is hard to pull off, as many of the films in this thread show by their failure, but this film is a reminder of the great joys that can be achieved when one does it well and with proper gusto. Highly recommended.

I Met Him in Paris (Wesley Ruggles 1937) Thoroughly average screwball with the typical dumb love triangle (more like a rectangle by the time Claudette Colbert gets done with her scorched earth policy towards suitors near the end), but the film is just barely funny enough to make it all work. I can't exactly recommend the film as a whole but Melvyn Douglas has two of the best lines I've ever heard in any film that more than justify seeing the film just to enjoy them in context, with the caveat that the surrounding film is just okay.

the Mad Miss Manton (Leigh Jason 1938) Infuriatingly dumb murder mystery with Barbara Stanwyck's daffy society dame running afoul of a killer while newspaperman Henry Fonda alternately damns and courts her. To state the obvious, these two stars fare much better a few years later in the Lady Eve. Unfunny and routinely idiotic, with the only saving grace being a brief plot-point centered on the construction of the NYC subway lines.

Once Upon a Honeymoon (Leo McCarey 1942) Wartime screwball that turns serious in the wake of what isn't very funny material about Ginger Rogers marrying an Austrian Nazi sympathizer. Cary Grant is bemused to the point of being a human cartoon as the radio personality turned spy pursuing Rogers, but he's the only thing that really works here.

She Married Her Boss (Gregory La Cava 1935) Melvyn Douglas marries his hardworking secretary Claudette Colbert and she proceeds to whip his personal life up in the same fashion as she previously did her work environment. Once you accept that this isn't really a comedy, much less a screwball one, it's a little easier to award some consolation points on its behalf, but this still isn't much of a movie.

Take a Letter, Darling (Mitchell Leisen 1942) Strident CEO Rosalind Russell hires Fred MacMurray to be her "personal secretary" and complications ensue. MacMurray's meek hesitancy is a nice switch from his usual blustery confidence, and Russell was literally put on earth to play roles like this. There's some mild fun had with the upending of gender norms and Russell's theatrics are always a treat, even if this ultimately doesn't add up to much.

Vivacious Lady (George Stevens 1938) Familiar tale of a stuffy guy falling for a loose woman and the social consequences wreaked in the wake of their marriage. Ginger Rogers is afforded some nice physical comedy in her raucous cat fight with her paramour James Stewart's former fiance, and Charles Coburn as the sternly disapproving father is always a welcome presence. This one gets better as it goes along, improving two-fold with each subsequent act, and it merits a slight recommendation.

No film subgenre is harder to marathon watch than this one-- I only have a few more films to go before I top off the initial 57 but I can't exactly say I'm itching to get them out of the way!