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?
Honestly, the Federation itself. I always thought you could make a really interesting show that was a mix of Federation politics and Starfleet intrigue; maybe a show about the Federation government on Earth that delved into Federation society and dealt with an overarching plot. Like maybe a season where a Federation colony at the edge of Federation space gets into conflict with another race/government, and for a season you get a mix of stories from the colony, a Starfleet ship sent to the colony to support, and the Federation Council on Earth.
Federation politics could be really interesting, but the shows always paid kind of disappointing lip service to it. One key example is in the DS9 episode “Rapture” when Bajor is being accepted into the Federation; the fact that a Starfleet admiral rather than a civilian Federation representative was overseeing the proceeding was lazy to me. Plus the entire ceremony was tiny and anticlimactic (and obviously barely got started since Sisko interrupted). Great potential for some real insight into the government but instead it’s just some rando admiral.
Yeah I came here to say the same thing.
That being said, I think it's a risk because any depiction of Federation life might be very close to current reality and thereby undermine a lot of the utopia they currently only allude to. Una's recent trial, for example, very much felt like a contemporary courtroom drama.
What I would like to see is more screen time for non-Starfleet agencies. Bring on the Federation diplomatic corps or, as you say, the relationship between Starfleet and its civilian leadership.