Basic Trope: The hat maker who doesn't wear his species' hat isn't taken seriously.
Straight: Volkas is a medical doctor on a Planet of Hats full of warriors. Without her help many wouldn't be able to recover from fights in order to learn to become better warriors, but they resent her for not being a warrior.
Exaggerated: Throngs of hecklers gather outside Volkas' clinic and regularly pelt her with produce when out in public.
Downplayed: Warriors acknowledge her contribution to their society, but she's still considered lower on the social ladder.
Her society believes in literal and social Darwinism, so "unnaturally" extending someone's life through medicine coddles the weak; which they despise.
Everyone in her society can choose to be a warrior in addition to their normal job, which is very common and sort of like playing recreational sports. So the scorn against Volkas who refuses to be a warrior in addition to a doctor is justified.
Volkas' society has Blue and Orange Morality, and the less her job tangibly contributes to society the more she's respected.
While the warrior's (the majority in control of her society) have no ill will towards doctors, Volkas and other doctors have no respect for warriors who go around getting themselves hurt on purpose. Essentially, My Species Doth Protest Too Much.
There are so few doctors in Volkas' society that doctors are almost worshipped.
The warriors and society in general look down on Volkas, the only physician around, and continuously say she has no honor or worth to society... Because she's an alcoholic, and most of her patients leave worse than how they came. If she were to sober up she'd be much more respected.
Volkas certainly acts as though she's undervalued by a Proud Warrior Race, and feels very unapreciated... Which is all in her head. Every sensible warrior knows they may some day need her services, and they hold her in high regard. Volkas merely has a fragile ego. Her patients constantly try to assure her she's an important part of their society.
Volkas is heckled for being a doctor...because she's bad at her job, and the heckler is saying that she'd probably be better at inflicting wounds, since she's crap at mending them. The much more competent Nerzan is lauded for his work.
Double Subverted: ... which is all a charade by society at large, who do it as a cruel game that nonetheless keeps the underclass happy. If Volkas ever tried to actually change things, they'd kill her without a second thought.
Parodied: Volkas' patients do nothing but complain about how useless she is even as they lay dying and her help is the only thing that will keep them alive.
Zig Zagged: At first it seems no one respects Volkas, but it's actually deferential silence, only it's actually just everyone holding in their giggles at fooling Volkas into thinking she's respected.
Averted: The Planet of Hats views Volkas and all other doctors as a profession that is neither honorable nor dishonorable.
Enforced: The author wants to add nuance to the Planet of Hats, so he introduces an underclass that is necessary but despised.
Invoked: An enemy spy who wants to weaken resistance stirs up prejudice against Volkas in the hope she'll get run out of town.
Exploited: Volkas uses this trope to threaten warriors with the dreaded "friends with a nerd" social stigma by acting like they studied medicine together.
Defied: The elders of the society know they need strong warriors, and without Volkas' and other doctors, they'll be worse off. So they make it a point to punish anyone who won't at least accept Volkas.
Discussed: "The worse part is they actually blame me for keeping them out of the field of battle while they recover. As if I have nothing better to do than keep idiots from jabbing each other with swords!"
Conversed: "Poor Volkas. It's a wonder this society hasn't imploded from stupidity."
Implied: Warriors of Volkas' race speak hyperbolically about how sucky it would be to be anything but a warrior, then they have a hearty and knowing chuckle.
Because of the deep disdain for non-warriors, fewer and fewer people become doctors, engineers and scientists. Eventually this leads to the best and brightest in society getting killed because they aren't talented as warriors, and then social and physical infrastructure collapses.
The despair at her situation forces Volkas and others to take a page from Ayn Rand and simply get up and leave. The warriors initially laugh off their absence until they start dropping like flies.
...And in the resulting anarchy only strong warriors survive, which leads to the birth of a "new" society that values skill as a warrior, while stigmatizing the "worthless" non-warriors who let the previous one collapse.
Volkas gets fed up with her mistreatment, and organizes a boycott of all non-essential services by all doctors, scientists, masons, etc. The resulting shortages gives the warriors an epiphany and ensures them Vetinari Job Security in the future.
Played For Laughs: Volkas is The Pollyanna, forever optimistic that she will be recognized for her efforts all while every one of her patients, family, and romantic interests treat her like The Scrappy.
Played For Drama: Because of all the rejection and cynicism of her society, Volkas decides to take up a sword and challenges an impossibly skilled warrior just so she can at least die honorably. Everyone thinks its a laugh until the day of the duel, when Volkas's opponent refuses to land the killing blow because she saved his life. Everyone is moved... but without a doctor on hand, Volkas dies due to blood loss, leaving everyone present the grim realization they realized her value too late.