Considering the small size of the Starfleet at present, I thought it would be a good idea for us to keep a record of what ships there are, and who their captain's are. Any that appear in your bios, post them here to be added.

NX-class ships:
Enterprise NX-01 - Captain Jonathan Archer
Columbia NX-02 - Captain Erika Hernandez
Challenger NX-03 - Captain ???
Discovery NX-04 - Captain Christina Curtis
Atlantis NX-05 - Under Construction
Endeavour NX-06 - Under Construction

NV-class (Intrepid-type) ships:
Intrepid NV-01 - Captain Carlos Ramirez
Republic NV-02 - Captain Jennings
Voyager NV-03 - Captain ???
Apollo NV-04 - Captain ???
Shenandoah NV-?? - Captain ???
Eagle NV-08 - Captain ???

NY-class (Neptune-type) ships:
Neptune NY-01 - Captain ???
Constellation NY-03 - Captain ???
Saratoga NY-?? - Captain Brody

I created a couple for Leo's bio:

Eagle (NV-08), and
Constellation (NY-03)

As the Discovery's resident alien, allow me to present the Vulcan ships that were named on the show. For brevity I'm only going to list the ships that are likely to still be in active service with the Vulcan Space Program.

Maymora-class (D'Kyr type) ships:
Maymora - Captain ???
Yarahla - Captain Tok
D’Kyr - Captain ???
Tal’Kir - Captain ???


Suurok-class ships:
Suurok - Captain ???
Ni'Var - Captain Sopek
Sh'Raan - Captain ???
Ti'Mur - Captain Vanik
Nyran - Captain Tavek

Being on an Andorian ship at present,

Andorian Imperial Guard Vessels:
AIG Docona - Commander Ga'tatyse th'Ravisee