I'm a huge fan of this topic, so I'll be pretty detailed. The short version is: I mostly wear Hanes/Fruit of the boxer briefs, sometimes regular boxers. I've noticed my peers do the same.
Here's the longer version...
I'm in the US, in the south. I mostly wear boxer briefs, but sometimes a pair of boxers.
Most of my boxer briefs are Hanes brand. Other brands I wear are Pair of Thieves, Fruit of the Loom, Jockey, Goodfellow (Target brand), Izod, Volcum, Reebok, Adidas, and Polo.
My boxers are either Hanes or Fruit of the Loom.
My peers wear pretty much the same thing, with the majority being boxer briefs. I will pretty much be just talking about people I know on some level, or come into contact with enough that I see their underwear. Meaning them bending over and it being exposed, going to the gym, going on trips and staying in hotels, that kind of stuff.
The boxer brief brands I've noticed my friends and other guys in my age group (I'm 33) wear are pretty much all of the ones I wear in addition to: Nautica, American Eagle, Tommy Hilfiger, Active (Walmart brand), Calvin Klein, Puma, Spyder, George (Walmart brand), Ethika, and PSD.
Most boxers don't have a name on their waistband, my best friend wears mostly Hanes and George boxers for what it's worth.
Is the underwear style driven/influenced by culture, country, ethnicity, society etc? ... Yes, I guess? I'd say I mostly see black guys wearing PSD or Ethika brand, same with Polo. Mostly white guys in something like American Eagle. Hispanic guys in a lot of Puma... Everything else is a mixed bag. It's hard to say what's worn in other countries, and how America differs. I'd guess from media. In my experience, most guys don't care too much about underwear. Or I should say, the guys I'm around.
I think it's mostly influenced by what's sold in stores. As an example entry level designer brand (Nautica, Izod, things like that) and athletic brand (Nike, Puma) underwear is really heavily stocked in places like Ross and TJ Maxx (Discount department stores... I guess?). I think that has brought them up in popularity.
Are there any particular underwear styles considered inappropriate or are laughed upon in your region? Inappropriate? No? Rarely are you seen in public in just you're underwear lol. Maybe like a thong or something. I can't remember the last time I saw someone in a pair of briefs in my age bracket, or even a few years younger. I think we're a little too old to make fun of someone's underwear, but for sure some "tighty whities".
what are dynamics between men & women's underwear styles in your countries, like one has to be necessarily shorter than others? I'm not 100% sure what you're asking. They make "boy shorts" that women wear. Kinda like trunks.
Do most men in your country feel comfortable in wearing small underwears like briefs / bikini briefs or are shy about it? Hard to say because I don't know anyone that wear briefs. I don't even see them in the locker rooms. Maybe like older men (60+) and even then, the ones I see are in boxers.
Keep in mind all of my answers are just based on people I come in contact with or people I know. I'm sure it will differ from different walks of life.