Frederick sure is observant. And by observant, we mean bordering on stalkerish. This guy really takes note of people's clothing—especially everything that Madame Arnoux wears.
Remember when Frederick first sees Madame Arnoux? The first thing he notices is what she is wearing:
She wore a wide straw hat with red ribbons which fluttered in the wind behind her. Her black tresses, twining around the edges of her large brows, descended very low, and seemed amorously to press the oval of her face. Her robe of light muslin spotted with green spread out in numerous folds. She was in the act of embroidering something; and her straight nose, her chin, her entire person was cut out on the background of the luminous air and the blue sky. (1.1.24)
Every detail of her being is burned into his brain and her clothing actually becomes part of his obsession with her. A few lines later, as he begins to wonder about this fascinating woman, we get this description: "He longed to become familiar with the furniture of her apartment, all the dresses that she had worn" (1.1.26). Becoming familiar with her dresses? How peculiar.
Just after he's left her for the first time, he's still dwelling on the details of her appearance:
Under the lowest flounce of her gown, her foot showed itself encased in a dainty silk boot of maroon shade. The awning made of ticking formed a wide canopy over her head, and the little red tassels of the edging kept perpetually trembling in the breeze. (1.1.64)
Madame Arnoux's clothes are a source of deep arousal for Frederick and almost seem more important than who she is on the inside, as they say. And of course, after an entire book about how obsessed he is with her looks, he rejects her at the end for having white hair. So yeah, we're going to go ahead and call him shallow.
P.S. Clothes can tell us a lot about the class someone is in. Remember how Frederick doesn't want to take Deslauriers anywhere because he looks too shabby with his frock-coat and dirty spectacles? Or how Frederick always makes a big deal about what he wears, especially when he is going to a party over at the Dambreuses? Yeah, this guy spends a lot on his clothes and other dandyish items. After all, he's got to keep up with everyone else.
The Ties that Bind
One item of clothing in particular pops up all over the place: ribbons. In fact, one of the first items of clothing Frederick notices on Madame Arnoux is the red ribbons on her wide straw hat. This moment actually anticipates the end of the novel, as she picks up her hat by the ribbons, just moments after the Seine riverbanks are compared to ribbons:
At last, the vessel set out; and the two banks of the river, stocked with warehouses, timber-yards, and manufactories, opened out like two huge ribbons being unrolled. (1.1.3)
But it's not just Madame Arnoux who sports them. Men and women alike wear 'em—notably, Rosanette and Louise Roque, who have them in their bonnets or curls or dresses. More than femininity, they're a sign of being well put-together. After a good party, Rosanette's house is covered with ribbons.
So what's with all the ribbons? One possibility: they might just symbolize how people in Sentimental Education are bound together—in loose and pretty ways. It's pretty clear that a lot about the relationships in the novel is surface-level.