NC = Non-Canon, SL = Ships of the Line, MA = Memory Alpha, MB = Memory Beta, BC = Bridge Commander
REG#      Name (Class [Where Found])
Experimental:
NX-01         Enterprise
NX-01A         Dauntless [VOY]
NX-02         Columbia [ENT]
NX-1000-F      Bonaventure [NC]
NX-2000         Excelsior [ST:III]
NX-2783         Surpasser [NC]
NX-59650      Prometheus [VOY]
NX-62341      Norway [ST:FC, DS9]
NX-72074      Pathfinder [NC]
NX-74205      Defiant
NX-74600      Intrepid [DS9]
NX-76220      Steamrunner [ST:FC]
NX-79221      Sabre [ST:FC]
NX-82424-A      Scorpion [NC]
NX-90000      Excalibur [NC]
NX-90493      Victory [NC]
NX-91001      Online (Sovereign Refit [NC])
NX-91809      Vivace [NC]
NX-95024      Valiant [NC]
NX-99000*      Sentinel [NC]
Starships:
NCC-C1200      Aurora (Cruiser [TOS])
NCC-F1913      Huron (Unknown [TAS])
NCC-S3700      Kobayashi Maru (Neutronic Fuel Carrier [ST:II])
NCC-173         Essex (Daedalus [TNG])
NCC-176         Horizon (Daedalus [DS9])
NCC-278-A      Akula (Akula [NC])
NCC-325         Woden (Antares [TOS])
NCC-330         Yorkshire (Antares [TOS])
NCC-403         Washington
NCC-501         Antares (Antares [TOS])
NCC-0514      Kelvin (Kelvin [ST:XI])
NCC-0582      Hargood (Kelvin Refit [NC])
NCC-585         Hermes (Hermes [ST:II])
NCC-595         Revere (Hermes [ST:TMP])
NCC-602         Oberth (Oberth [ST:VI])
NCC-621         Columbia (Hermes [ST:TMP])
NCC-623         Coppernicus (Oberth [ST:IV])
NCC-638         Grissom (Oberth [ST:III])
NCC-956         Eagle (Constitution [ST:VI])
NCC-1010      Akyazi (Akyazi [SL])
NCC-1017      Constellation (Constitution [TOS])
NCC-1017-A      Constellation (Constellation [NC])
NCC-1018      Eleusis (Akyazi [SL])
NCC-1024      Whorfin (Unknown [ST:VI])
NCC-1048      Arbiter (Akyazi [SL])
NCC-1048-A*      Arbiter (Sentinel [NC])
NCC-1062      Talence (Arbiter [SL])
NCC-1090      Akula (Akyazi [SL])
NCC-1098      Sjoto (Akula [SL])
NCC-1175      Chester (Constitution [NC])
NCC-1371      Republic (Constitution [TOS])
NCC-1543      Fontana (Constitution [NC])
NCC-1598      Scovil (Unknown [ST:VI])
NCC-1621      Mayflower (Miranda [ST:XI])
NCC-1631      Intrepid (Constitution [TOS])
NCC-1647      Farragut (Constitution [TOS])
NCC-1657      Potemkin (Constitution [TOS])
NCC-1664      Excalibur (Constitution [TOS])
NCC-1672      Exeter (Constitution [TOS])
NCC-1692      Helin (Oberth [ST:VI])
NCC-1700      Constitution (Constitution [TOS])
NCC-1701-(A-J)   Enterprise (Various) NOTE: No H or I
NCC-1703      Hood (Constitution [TOS])
NCC-1709      Lexington (Constitution [TOS])
NCC-1717      Yorktown (Constitution [TOS])
NCC-1727      Newton (Unknown [ST:XI])
NCC-1764      Defiant (Constitution [TOS])
NCC-1769      Armstrong (Unknown [ST:XI])
NCC-1776      Patriot (Patriot [NC])
NCC-1830      Miranda (Miranda [TNG])
NCC-1837      Lantree (Miranda [TNG])
NCC-1864      Reliant (Miranda [ST:II])
NCC-1894      Sagittarius (Archer [NC])
NCC-1940      Unknown (Soyuz [TNG])
NCC-1941      Bozeman (Miranda [NC])
NCC-1963      Springfield (Oberth [ST:VI])
NCC-1974      Constellation (Constellation [TNG])
NCC-1985      Armed and Dangerous (Proxima [NC])
NCC-2000      Excelsior (Excelsior [ST:VI])
NCC-2053      Valorous (Excelsior [MB])
NCC-2060      Kelly (Excelsior [NC])
NCC-2250      Franz Ferdinand (Franz Ferdinand [NC])
NCC-2265      Atraiu (Atraiu [NC])
NCC-2544      Repulse (Excelsior [TNG])
NCC-2573      Roosevelt (Excelsior [VOY])
NCC-2582      Farragut (Excelsior [DS9])
NCC-2590      Valkyrie (Constellation [TNG])
NCC-2593      Hathaway (Constellation [TNG])
NCC-2713-D      Arcadia (Century Refit {NC])
NCC-2784      Oklahoma (Surpasser [NC])
NCC-2785      Albany (Surpasser [NC])
NCC-2800      Constellation (Constellation [TNG])
NCC-2893      Stargazer (Constellation [TNG])
NCC-3069      Magellan (Constellation [TNG])
NCC-3549      Niijima (Niijima [NC])
NCC-3737      Oklahoma (Unknown [NC])
NCC-3737-A      Oklahoma (Okalahoma [NC])
NCC-3890      Gettysburg (Constellation [TNG])
NCC-4702      Sundown (Miranda variant [NC])
NCC-5300      Einherjer (Einherjer [NC])
NCC-7100      Serenity (Constellation [NC])
NCC-9754      Victory  (Constellation [NC])
NCC-9761      Gihlan (Constellation [TNG])
NCC-10521      Ambassador (Ambassador [TNG])
NCC-10532      Horatio (Ambassador [TNG])
NCC-11965      Hawking (Oberth-Pegasus [NC])
NCC-13958      Okinawa (Excelsior [DS9])
NCC-14232      Berlin (Excelsior [TNG])
NCC-14811      Hussatonic (Oberth-Pegasus [NC])
NCC-14932      Tecumseh (Excelsior [DS9])
NCC-14958      Fearless (Excelsior [TNG])
NCC-18253      Potemkin (Excelsior [TNG])
NCC-19002      Yosemite (Oberth [TNG])
NCC-21166      Brattain (Miranda [TNG])
NCC-26136      Zhukov (Ambassador [BC])
NCC-26198      Valdemar (Ambassador [DS9])
NCC-26510      Yamaguchi (Ambassador [DS9])
NCC-26517      Excalibur (Ambassador [TNG])
NCC-26531      Exeter (Ambassador [DS9])
NCC-26610*      Unknown (Hobbs Son [NC])
NCC-26632      Gandhi (Ambassador [TNG])
NCC-26849      Adelphi (Ambassador [TNG])
NCC-31800      Bonestell (Oberth-Pegasus [NC])
NCC-31859      Sitak (Miranda [NC])
NCC-31905      Shirkahr (Miranda [NC])
NCC-31911      Saratoga (Miranda Variant [DS9])
NCC-32628      Caldos (Excelsior [MB])
NCC-34099      Livingston (Excelsior [DS9])
NCC-38995      Crockett (Excelsior [DS9])
NCC-38997      Malinche (Excelsior [DS9])
NCC-39412      Unknown (Centaur [DS9])
NCC-40512      Gorkon (Excelsior [TNG])
NCC-41069      Kolno (Excelsior [MB])
NCC-41987      Orlando (Constitution Refit [NC])
NCC-42043      Centaur (Centaur [DS9])
NCC-42111      Frederickson (Excelsior [DS9])
NCC-42136      Cairo (Excelsior [TNG])
NCC-42285      Charleston (Excelsior [TNG])
NCC-42296      Hood (Excelsior [TNG])
NCC-42768      Lakota (Excelsior Refit [DS9])
NCC-42857      Grissom (Excelsior [TNG])
NCC-43305      Valley Forge (Excelsior [DS9])
NCC-46200      Blackeagle (Excelsior [MB])
NCC-47936      Vasidrun (Excelsior [NC])
NCC-50000      Cheyenne (Cheyenne [TNG])
NCC-50331      Biko (Oberth [TNG])
NCC-50446      Crazy Horse (Excelsior [TNG])
NCC-51905      Edmonton (Alberta [NC])
NCC-52136      Appalachia (Steamrunner [DS9])
NCC-53847      Pegasus (Oberth-Pegasus [TNG])
NCC-53911      Tsiolkovsky (Oberth Civilian [TNG])
NCC-57300      Chekov (Springfield [TNG])
NCC-57924      Apache (Excelsior [MB])
NCC-58314      Tyr (Excelsior [NC])
NCC-58800      Niagara (Niagara [TNG])
NCC-58804      Princeton (Niagara [TNG])
NCC-58920      Olympic (Olympic? [TNG])
NCC-59318      Cochrane (Oberth [TNG])
NCC-59983      Raman (Oberth [TNG])
NCC-60202      Nebula (Nebula [MB])
NCC-60205      Honshu (Nebula [DS9])
NCC-60597      Farragut (Nebula [ST:GEN])
NCC-60805      Nightingale (Nebula [BC])
NCC-61826      Monitor (Nebula [TNG])
NCC-61827      Merrimac (Nebula [TNG])
NCC-61832      Lexington (Nebula [DS9])
NCC-61947      Yeager (Saber [ST:FC])
NCC-61952      Proxima (Nebula [DS9])
NCC-62006      Hera (Nebula [TNG])
NCC-62043      Melbourne (Nebula Variant [TNG])
NCC-62048      Bellerophon (Nebula [DS9])
NCC-62100      T'Kumbra (Nebula [DS9])
NCC-62136      Zuhkov/Hammarskjold (Ambassador [TNG])
NCC-63185      Sunchaser (Excelsior [MB])
NCC-63552      Agamemnon (Akira [NC])
NCC-63293      Rabin (Akira [ST:FC])
NCC-63549      Thunderchild (Akira [ST:FC])
NCC-63646      Akira? (Akira [ST:FC])
NCC-63744      Bolsetu (Excelsior [NC])
NCC-64088      Devore (Akira [BC])
NCC-64720      Berkely (Nebula [BC])
NCC-65342      Unknown (New Orleans [TNG])
NCC-65410      Unknown (Nebula [TNG])
NCC-65420      Phoenix (Nebula [TNG])
NCC-65426      Garuda (Nebula [DS9])
NCC-65491      Kyushu (New Orleans [TNG])
NCC-65549      Specter (Akira [ST:FC])
NCC-66613      Khitomer (Nebula [BC])
NCC-66808      Ulysses (Nebula [DS9])
NCC-68500      Shelby (Cobra [NC])
NCC-69302      Geronimo (Akira [BC])
NCC-69480      San Francisco (Galaxy [BC])
NCC-69903      Osiris (Nebula [MB])
NCC-70258      Denali (Nebula [MB])
NCC-70352      Leeds (Nebula [TNG])
NCC-70637      Galaxy (Galaxy [DS9])
NCC-70915      Bonchune (Nebula [VOY])]
NCC-70782      Olympia (Nebula Variant [TNG])
NCC-71099      Challenger (Galaxy Variant [NC])
NCC-71201      Prometheus (Nebula [DS9])
NCC-71257      Krakow (Krakow [NC])
NCC-71805      Endeavour (Nebula [ST:FC])
NCC-71807      Yamato (Galaxy [TNG])
NCC-71820      Magellan (Galaxy [DS9])
NCC-71832      Odyssey (Galaxy [DS9])
NCC-71846      Maddison (Galaxy [ST:FC])
NCC-71854      Venture (Galaxy [DS9])
NCC-71867      Trinculo (Galaxy [DS9])
NCC-71872      Levski (Proxima [NC])
NCC-71879      Dauntless (Galaxy [BC])
NCC-71928      Danysz (Sobieski [NC])
NCC-72015      Sutherland (Nebula [TNG])
NCC-72380      Nova (Nova [VOY])
NCC-72381      Equinox (Nova [VOY])
NCC-72701      Rhode Island (Nova [VOY])
NCC-73209      Paccia (Defiant [NC])
NCC-73515      Nova (Nova [ST:NEM])
NCC-73598      Titan (Galaxy EVO [NC])
NCC-73620      Ahwahnee (Cheyenne [NC])
NCC-73811      Sovereign (Sovereign [BC])
NCC-73865      Bozeman (Sovereign [ST:FC])
NCC-74121      Elkins (Intrepid Variant [DS9])
NCC-74206      Gallant (Defiant [MB])
NCC-74210      Valiant (Defiant [DS9])
NCC-74213      Thunderer (Defiant [NC])
NCC-74215      Bannockburn (Defiant [NC])
NCC-74217      Ironheart (Defiant [NC])
NCC-74295      Regal (Sovereign [NC])
NCC-74446      Imperial (Sovereign [NC])
NCC-74432      Integra (Nova Variant [NC])
NCC-74569      Destiny (Sovereign [DS9])
NCC-74656      Voyager (Intrepid [VOY])
NCC-74656-A      Voyager (Vivace [NC])
NCC-74699      Antares (Defiant [NC])
NCC-74705      Ballerophon (Intrepid [DS9])
NCC-74733      Avalon (Intrepid [MB])
NCC-74780      San Francisco (Galaxy [NC])
NCC-75058      Sentinel (Sovereign War Refit [NC])
NCC-75120      Pharoah (Sovereign [NC])
NCC-75418      Valiant (Defiant [ST:NEM])
NCC-75420      Lynx (Defiant [NC])
NCC-75633      S?o Paulo (Defiant [DS9]) (No Longer in Use)
NCC-75639      Athena (Sun Tzu [NC])
NCC-75952      Sultan (Sovereign [NC])
NCC-76890      Patriot (Aegis [NC])
NCC-77502      Sovereign (Sovereign [ST:FC])
NCC-77777      Sam (Proxima [NC])
NCC-78135      Dakuwaqa (Tiburon [NC])
NCC-78201      Calgary (Intrepid [NC])
NCC-78562      Pegasus (Galaxy [NC])
NCC-78566      Olympus (Galaxy [NC])
NCC-79099      Emperor (Sovereign [NC])
NCC-79243      Osaka (Intrepid [NC])
NCC-79548      Wolverine (Defiant [NC])
NCC-80060      Supreme (Sovereign [NC])
NCC-80132      Legend (Sovereign War Refit [NC])
NCC-81339      Lionheart (Arsenal [NC])
NCC-81372      Kyoto (Intrepid [NC])
NCC-81729      Mako (Intrepid [NC])
NCC-82104      Alexandria (Intrepid [NC])
NCC-82322      Supremus (Supremus [NC])
NCC-82501      Australia (Commonwealth [NC])
NCC-85501      Kyoto (Kyoto [NC])
NCC-91810      Schumann (Vivace [NC])
NCC-91815      Chopin (Vivace [NC])
NCC-92792      Regency (Krakow [NC])
NCC-93284      Ticonderosa (Aquiescent [NC])
NCC-94893      Atlantic (Excalibur [NC])
NCC-96663      Fuoco (Vivace Variant [NC])
NCC-99100*      Dakeen (Sentinel [NC])
NCC-99200*      Calypso (Sentinel [NC])
NCC-99330*      Khitomer (Sentinel-Amherst [NC])
NCC-99450*      Nefertiti (Sentinel-Amherst [NC])
NCC-122378      Lennox (Achilles [NC])
NCC-954266*      Unknown (Hobbs Son [NC])
Auxiliary Craft:
3400         Galileo (Shuttlecraft [TOS/ST:V])
CS-I04         Clydesdale (Shuttlepod/Yard Tug [ST:IV])
FV500N         Cochrane (Long Range Scoutcraft [TAS])
NAR-18834      Vico (Oberth [TNG])
NAR-25820      Sydney (Exec Shuttle [ST:VI])
NAR-32450      Raven (Civilian Research Vessel [VOY])
NCC-S2000        Sydney (Civilian Transport Vessel [ST:III])
NCC-72450      Danube/Yellowstone (Runabout [TNG/DS9])
NCC-72905        Orinoco (Runabout [DS9])
NCC-74656      Delta Flyer
NCC-75227      Venture (Unknown [ST:9])
PS-D1         Caretaker (Passenger Shuttle [ST:TMP])
SC-4         Janeway (Shuttlecraft [VOY])
Inconsistencies:
NCC-62136 is known by canon references as both the Zuhkov and the Hammarskjold.  No explanation is given.
NOTE: The above does not contain any data from the technical manuals, novels or the Star Trek Encyclopedia.