What are scientists of the Star Trek universe working on if anything?

[Mojochi]

Holding people in the transporter buffer during interstellar travel.

Think of all the trouble that would be avoided if passagers were beamed up only as far as the pattern buffer, held there during the journey, then beamed down to their destination.

Never actually setting foot on the ship.

As well as appling that tech to medical emergency situations.
Instances where life saving aid could not be offered in time to save a life would have them instituting a transport holding pattern until said ministrations are available

Artificial wormnholes would seem to be a worthwhile endeavor in an age where warp travel causes damage to space

Duplicating artificial intelligent life seems to be a big interest, either holographic or robotic

 

[matthunter]

As well as appling that tech to medical emergency situations.
Instances where life saving aid could not be offered in time to save a life would have them instituting a transport holding pattern until said ministrations are available

Medical stasis is a thing, as of the TNG ep where the crew de-evolve (Beverley is placed in it after Worf sprays her with neurotoxin).

Artificial wormnholes would seem to be a worthwhile endeavor in an age where warp travel causes damage to space

DS9 ep Rejoined has the Federation working on this.

Duplicating artificial intelligent life seems to be a big interest, either holographic or robotic

Maddox' interest in Data and Lal agrees. The TNG novel "Immortal Coil" studies Data in the context of the androids and AIs we saw in TOS through the ongoing work of Flint (revealed to have faked his death) and the Exo III androids who are after a new prototype "Holotronic" android. This is later picked up in the post-Nemesis novels beginning with The Cold Equations.

 

[Velocity]

and just think of all the history there would be to study!

 

[Voth commando1]

Better medical technologies and procedures-DS9 and Voyager

Artificial wormholes-DS9

Better FTL travel-Voyager, DS9, TNG, ENT

Computer science-TOS, VOY, TNG

Robotics-TNG

Creating a continent in the Atlantic-TNG

Terraforming-TNG

There are quite a few episodes throughout Trek where the development of new technologies or improving old ones is discussed.

Not to mention "non-technological" sciences such as astrophysics, biology, archaeology, etc... That are referenced throughout Trek.

I imagine the Federation is awash with scientists in every field.