Sadly, I am an extremely picky eater, so it's good for me to know how to tell people this to explain my selectivity with restaurants and meals.
But I'm a bit confused by the phrase "faire la fine bouche," because I'm not familiar with the word fine in French.
What does fine mean in this expression? I couldn't find any definition that made sense-- Wiktionnaire lists it as a typography term and secondarily as something to do with cognac.
"Fine bouche" means "thin mouth" (the prononciation is close in french and in english, by the way). It was originally "petite bouche" ("small mouth"). If you're a gourmet, you're more likely to eat your food little by little, not opening your mouth too wide and less likely to swallow big chunk without masticating. Thus the expression.
Like Papa Poule said in the comment, you should be careful with this expression : it imply that what was served was not good enough for you. The chef or host could ressent you. You could use the litteral translation for "I'm a picky-eater" ("Je suis un mangeur exigeant/difficile/pointilleux") for added politeness.
Fine bouche is literally meaning: Fine mouth or Delicate mouth. It's an expression that relates to the mouth you use to eat and the fact that people have to be delicate in their food choice: you are a picky eater, hence not everything will go through this mouth. You have this the expression like this.
Fine in this case is linked to the quality of the meal you are expecting. You can find epiceries fines or boucherie fine still in some places. There, you'll find high quality or rare products.
Faire la fine bouche
Means been difficult and needing special treatment and better meals
Avoir un palais fin
Means having good and precise taste.
It's also linked to un travail fin which means that it's a work of high quality and high precision.
