One of the fascinating things about this "third generation" of Star Trek (starting either with Star Trek 2009 or with Discovery) is the way the Star Trek universe has started to knit itself closer together by referencing existing backstory. For example, Discovery wholeheartedly embraced the idea that Andorians and Tellarites are key Federation members and should therefore be highly visible in Starfleet, building on lore originally implied in TOS, largely ignored by TNG, DS9, and VGR, and re-embraced by ENT. Prodigy, for its part, leaned very heavily on VGR for its worldbuilding source material.
This has also produced some interesting quiet exclusions from recent stories -- not to suggest they've been "decanonized" or anything like that, but clearly have been deprioritized. The Tholians come to mind as a ready example of this. Like the Gorn, they debuted in TOS, received stray mentions in DS9, before making an on-screen return in ENT. I wonder if the SNW writers considered using the Tholians but balked at a villain that had such different atmospheric requirements, and all the consequences that entails in terms of dramatic presentation. The Denobulans also seem to fall into a similar bucket; outside of a pair of appearances in PRO, they have received nary a mention since ENT.
Then of course we have the lengthy list of "one-off" civilizations, including the likes of
-the Sheliak
-the Husnock
-the Tzenkethi
-the Jarada
-the Miradorn
And in terms of "underexplored corners", I've only been focusing on the civilizations, but there are any number of other corners we could poke into. The Department of Temporal Investigations, the Corps of Engineers, the Federation Council, the Lunar Colonies... the Trekverse is littered with these little crumbs all over the place -- tiny seeds of ideas that suggest opportunities for imagination.
For my part, I would love to learn more about the Sheliak. For one thing, they seem like they would benefit from the advances in CGI over the last 30 years. But I like that they seem equally matched to the Federation in terms of strength, and that their hyperfocus on legal compliance gives them a generally underused "hat" to wear in the Trekverse. They have some similarity to Vulcans, but taken to an extreme, and layered in with real disdain for "lower life forms" that I think would make for a fascinating "adversary" -- I'd love to see Captain Pike or Captain Seven in a verbal jousting match with a Sheliak commander.
What is an underexplored corner of Trek lore that you think merits exploration?
Starfleet personnel are a subset of the total human population. Off the top of my head I'm not aware of any canon figure about what percentage of the human population is in Starfleet, or the total number of Starfleet, but presumably it is not a huge percentage of the total.
There are numerous notable civilian characters in Trek, but in general the franchise presents us with a vision where, contrary to the in-universe reality, civilian life is sort of at the periphery of Starfleet. Civilians are people that Starfleet personnel find to be colorful characters, sometimes become romantically involved with, people who serve Starfleet personnel when they're on vacation or otherwise at leisure, sometimes criminals, etc.
The notable civilian characters we see on screen tend to have their entire lives circumscribed by Starfleet -- think Jake Sisko or Keiko O'Brien.
I'd be really curious to see not just a human civilian character but a civilian world fully developed over the course of a series. What is it like to be "just some person" living their life?
I suspect this isn’t covered for a variety of reasons, but what we do see shows an Earth steeped in nostalgia blending with technology that alleviates the overwhelming majority of material concerns. No one needs to worry about food, shelter, clothing, or profit. They work to better themselves for its own sake and they relax and stuff.
I think the message is that Earth is boring and humans, having created paradise, left it for knowledge of the stars. Civilian life is coffee and breakfast in the morning before dropping the kids off at school, plenty of time to pursue niche hobbies, and probably also time to pursue some career interest and education.
However, you raise an interesting point in that civilians from the perspective of Starfleet officers are even more out there. These are folks who often wanted to go even further than the rest of society wanted to go. Fringe people with eccentricities to un civilized to stay within the Federation. The intersection of these two kinds of civilians is what I would like to explore.
A happy family who have lived in paradise are suddenly Swiss Family Robinson or Lost in Space style thrust into deep space and must survive without the comforts of paradise would be an interesting way to tell the story of what it’s like to be a civilian in the Federation.