I don't particularly like to
fly the Falcon, I will admit. I like symmetry, and with the Falcon it's hard to have a loadout that's both good and symmetrical. It absolutely should have an ion beam, but two is too much. It bothers me a lot less when it's an AI officer taking the wheel, though, in which case - well, it's the archetypical light cruiser. Cheap, sturdier and more powerful than a destroyer, capable of outrunning most things it can't outgun. It's a bit weak to fighters, though.
In terms of carriers... I strongly prefer battlecarriers, for which I consider the Heron and the Legion to be the only qualified examples. The Mora is too slow, more of a hulking presence that catches unaware opponents with its missiles than a real battlecarrier. I've seen the Drover described as such, but I don't agree with that description at all. The Drover is as pure a carrier as you can get.
Early-game I do take a surprising amount of joy out of piloting a Gemini or Drover and supporting the rest of my fleet with Thunders. (Thunders, incidentally, are probably my favourite fighter to use, and if I'm not piloting a carrier myself I will make sure to have at least one Thunder-carrier on the battlefield so I can throw them at hot-spots.) Personally I find the Heron too undergunned to fly and I always forget to use the ship system, which can be fiddly and flux-intensive with the timing. It's also not really got a lot of OP. It's quite good in AI hands though I prefer the Drover for pure carrier capability. The Legion is really fun, but I don't feel I can justify filling four hangar bays with Thunders. Instead I often end up going with the Onslaught.
Early on I often fly an Enforcer with a converted hangar bay and a Talon wing, instead. It works surprisingly well with even just the fleet-wide carrier skills, and the personal carrier skills end up making that lone wing count for a surprising lot.
Overall I prefer low-tech ships, occasionally dipping into mid-tech. For high-tech I usually limit myself to the Wolf and Tempest - I have a personal bias against most other high-tech ships, though I will occasionally fly the Medusa.
The Lasher used to be my favourite frigate in 0.72, but with the nerf to Unstable Injectors it's just not very survivable in 0.8. It's good early on, but it deals poorly with sustained pressure and it finds itself in an uncomfortable middle-ground in terms of speed, firepower and durability versus the Centurion and Brawler on one end and the Hound, Wolf and Tempest on the other. SO solves that in player hands but is often suicidal in AI hands. I still like it best in terms of its basic concept, but it is too slow to fly and very vulnerable against certain enemy types.
Instead I favour the Wolf and the Tempest. Fast, evasive, decent and flexible firepower.
My favourite cruiser is probably the Eagle. I like the Dominator in theory, but I find the firing arcs of its main guns a bit too restrictive compared with its low maneuverability. The Eagle is fast and versatile. You can really see why it's the most common cruiser in the sector. It just works in every situation. Its only downside is that you have to give up a lot of offensive power if you want to have more point defense than the average Destroyer.
My favourite capital ship (and favourite ship period) is far and away the Onslaught. If I were not a lazy butt, the XIV variant would be my avatar. The vast amount of ordnance points it has, the in-built TPCs, the excellent burst and sustained firepower, the supremely heavy armour, the way it scales up with combat skills, the heavy point defenses, even the ship system to some extent, and of course, the ship's looks and lore. It is a brick. It is the best of all bricks. Starsector lives and dies for me depending on how it treats the Onslaught. It's my baby, Alex, so don't you ever nerf it.
You have not lived until, with a Combat build, you've soaked up an entire ion pulser magazine to the side and laughed it off.
In mods, there are a lot of ships I like.
Between the non-faction Ships & Weapons Pack, Underworld, Disassemble & Reassemble and Expanded Arsenal, there is the Predator, Predator-X, Wren, Taurus, Manta, Vanguard and Duelist. The Predator is like a lower-tech Wolf in function, the Wren is cheap but very fast, the Taurus is tough and has flexible weapon mounts, the Manta is a destroyer-sized light battlecarrier of sorts, the Vanguard's main gun and ship system is very lovely, and the Duelist's dual-shield system is just fancy enough to win me over.
Most non-gimmick Imperial ships, period. They sometimes feel a bit too good compared to regular low-tech ships and the II is an enemy of the Hegemony whose Onslaught I love so much, though. Of course, Imperial ships are more-or-less intended to be all-round solid. A good mix of weapon types, sturdy, fast... a bit ordnance-point starved, though. Particular mentions go to the Princeps, Sebastos and Dominos, followed by the Decurion, Praetorian, Legionnaire and Dictator.
From Diable there is the Gust. I have not yet played around with the bigger ships but I imagine I would quite like both the Storm and Maelstrom.
From Dassault-Mikoyan Engineering, the Kormoran stands out. Overall I dislike the aesthetics even if I like the gameplay, alas.
From Shadowyards Reconstruction Authority, there is the Morningstar and the Charybdis. Both are flexible, maneuveral ships with drone/fighter support; in the latter case an excellent example of a combat carrier.
From Scy, there is very noteably the Corocotta. Competes with DA's Gust and II's Sebastos for being my favourite mod cruiser, certainly. The fragile combination of speed and firepower is exhilarating.
From Blackrock Driveyards, there is the ever-predictable Nevermore. I distinguish myself by not counting it as my absolute favourite, preferring instead the trio of cruisers mentioned previously. I also quite like the Knight, even if it is incredibly starved for ordnance points.
From the Templars - well, I quite like the Elucidator. The Bulwark is excellent in AI hands but a bit of a finicky snooze to fly personally. The Paladin I only have a limited amount of experience with, and never in a proper combined-arms setting, and I have no experience at all with the Archbishop, but I imagine I would quite like both of these.