Peggy (Molly Shannon) begins “Year of the Dog” with a serious case of puppy love. The puppy in question is a precious, perky little beagle named Pencil. He’s the most important thing in her life, though that doesn’t really say much: Peggy has few friends, and her daily routine involves coming home from her boring job, feeding Pencil, and settling down with him in front of the TV. (Sometimes she even rubs a little moisturizer on his paws while they watch.)

It may seem pathetic, but this relationship is more rewarding than any Peggy has with her own species, who view her company as little more than an opportunity to prattle on about something. Her boss (Josh Pais) complains about work, her best friend (Regina King) goes on about her beau, and her brother and his wife (Thomas McCarthy and Laura Dern) periodically invite her over just so they can hold forth on health risks to their children.

But then Pencil gets into something toxic in the neighbor’s garage, a tragedy that turns “Year of the Dog” into something resembling a romantic comedy — albeit one in which dogs compete with men for the leading lady’s interest. Peggy’s neighbor (John C. Reilly) comes by to express his condolences, then asks her out on a date. She also catches the eye of a friendly animal hospital employee named Newt (Peter Sarsgaard).

Newt sets Peggy up with Valentine, a German shepherd in need of a home. Big and ill-behaved (he was abused by his previous owner), Valentine is a challenge to love — unlike Newt, who dazzles Peggy with his kindness and knowledge of canine psychology. Under Newt’s tutelage, Peggy turns vegan and volunteers at a dog shelter. He’s her ideal mate — except he’s asexual.

The main problem with “Year of the Dog,” written and directed with droll wit by Mike White, is that the audience sees this roadblock, and others, coming a mile away. When Newt announces he’s celibate — and kind of prefers guys anyway — it doesn’t exactly arrive as a plot twist. But it sends Peggy reeling.

She channels her romantic frustration into activism, morphing into a militant animal-rights advocate, which makes her a thorn in the side of her boss (she brings petitions to work) and just about everyone else. Part of you cheers her on — she has turned the tables on all those people who used to talk her ear off, and gives a poignant explanation for why she’s a vegan: “It’s nice to have a word that can describe you.” But another part of you knows Peggy is just being silly, such as when she refers to the goings-on at her local poultry plant as a “holocaust” and starts secretly cutting checks to her favorite charities on her boss’s behalf.

Adamant and awkward, Peggy is the butt of many of the film’s jokes, some of which are very funny. But she’s a risky lead character—not so much because it’s hard to like her, but because it’s nearly impossible to respect her. She plots to kill her neighbor, whom she blames (unjustly) for Pencil’s death, and a desperate burst of compassion leads her to adopt a mob of stray dogs. Unsurprisingly, they tear her house to shreds.

A woman this hopelessly misguided can’t carry an entire movie, and neither, to judge from this performance, can Ms. Shannon. The “Saturday Night Live” graduate is a specialist in zany, cartoonish characters, but Peggy needs something — more complexity, or maybe just dignity — to elevate her from a mere object of pity. Ms. Shannon frowns and furrows her brow much more than usual, but doesn’t come up with what’s needed.

Mr. White, who has penned both conventional comedies (“Orange County,” “The School of Rock”) and offbeat ones, is a very good writer, even if his rancid breakout film “Chuck & Buck” earned undue critical praise. The delicate balance of twee and sardonic humor he achieves in “Year of the Dog,” his directorial debut, is perfectly suited to human-pet relations, and the film contains a small treasure trove of clever digs and apercus. The supporting cast is excellent — in particular Mr. Sarsgaard, who gives life to the film’s most understated character, a sensitive man at ease with his preference for animals.

Unfortunately, Peggy, who spends most of the film at ease with nothing, never really becomes anyone most viewers might recognize in themselves. You watch her coo and rant and you feel a bit sad for her, but you’re not really with her — and when, in one her more frazzled moments, Peggy puts a menagerie of baying dogs in her back yard, you’re with her neighbor 100%. Shut those things up!