My thoughts:
First, and foremost, I'm under the impression that not all ships need to be viable. The Condor is obviously an inferior, makeshift, carrier. Same with anything "Mk. II." Second, perfect parity among ships/classes isn't achievable so I'm not going to try. My biggest litmus test for all of these ships are "Do I use them, and if not, why?" Most support ships aren't listed because a transport or fuel hauler doesn't need to be balanced, IMO.
Frigates:
I consistently use Wolves, Lashers, Tempests, Centurions, Kite(A)s, and Omens. I generally don't use phase ships (or pilot them) but Afflictors are good. I have no experience with the Shade. The Monitor is fine, though I don't use it enough. Hounds are also where they need to be as expendable haulers.
Brawler - The Brawler is touted as a beefy patrol Frigate but while it has decent firepower, it's an anvil, not a hammer. The Centurion already occupies this role so I feel the the Brawler is redundant relative to the Centurion. I could live with its speed if were given Accelerated Ammo Feeders as a ship system. Whenever I have them in my fleet, they get flanked by faster frigates or die to fighters. My suggestion is bump speed by 10 and give it AAF. The TT version is in a weird spot because medium energy is just a different beast altogether. If Plasma Jets let you jump backwards, I think it'd be a decent hit-n-run platform but as it is, you jump in and get smashed.
Scarab - Suffers from "Small Energy Mount Syndrome." Outside of AM Blasters, it can't pressure shields well enough. With the changes to beams and hard flux, maybe it will find a new niche but as it stands, the Scarab is a fantastic concept with abysmal execution. If Small Energy mounts had more options, it'd be pretty cool. On the other hand, it has too many mounts (and the side mounts have terrible firing arcs) to fit with its OP/dissipation. I think the side mounts are a trap and could be removed altogether with no negative consequences. If that meant the center mount could be changed to a universal or another synergy (toying with a medium energy), I think the Scarab would have more versatility. It's a hit-n-run ship to be sure but even though it can avoid most fire in a 1v1 situation, even with Temporal Shell, it's not generating appreciable damage in the moments of opportunity it has. (This has been my complaint for the Shrike, as well). Finally, the ship is too dang rare. I never see them or get blueprints for them.
Vigilance - I don't think the Vigilance is bad, I just know that in battle its usefulness dwindles about 45 seconds in due to relying on missiles. It should have built-in Expanded Missile Racks, a la the Gryphon.
Wayfarer - Its initial premise was a decent combat hauler but like the Centurion prior to its arcs getting changed, it just couldn't bring enough firepower to bear to be worth it. It needs the Centurion treatment of having overlapping arcs. As a player ship, its frustrating. Any more than arc changes would be overkill though.
Destroyers
I frequently use Hammerheads, Sunders, Drovers, and Enforcers. They're not all equal but all are viable. I don't use Medusas but that comes down to playstyle not inadequacies of the ship. Condors are fine, as are Mules, for what they are and need to be.
Enforcer - They are currently the losers of the "Big 5" of combat Destroyers. Hammerheads are all-around better, Sunders and Medusas are more specialized, and Drovers are top-tier carriers. Enforcers are logistically more expensive and though they're bricks, they pretty much have to have two flak cannons because they don't have the dissipation for 5 medium ballistics. More often than not, Hammerheads can out DPS them due to AAF so that leaves the Enforcer in a spot where they should win the "more gun" duel but can't. My suggestion: Medium Ballistics Integration (similar to the Conquest). It doesn't fix dissipation per se but it allows the Enforcer to max its stats or hull mods as necessary. It should always be the biggest gun among the destroyers.
Shrike - "Small Energy Mount Syndrome". That's why I petitioned awhile back for the universal mount that, eventually, the (P) version got. It's not a bad ship and it doesn't die that often when I have it in my fleet but its role is limited to support. I usually throw an Ion Beam on it and let it go. I want it to be aggressive, though, and small mounts just aren't suited for it. A Heavy Blaster is also a pretty big strain on it. Perhaps I'm just trying to fit a square peg into a round hole. More OP wouldn't hurt.
Cruisers
Honestly, I have almost zero complaints with the Cruiser lineup. All have their niches, all have pros/cons, and some are better than others (*cough* Aurora) but all are good. The Gryphon is the one I use the least but I can't speak to it because I don't feel I have anything to add. I just don't like the idea of a ship that becomes mostly useless at some point in a battle. Only one thing frustrates me: the Apogee and the firing arcs of the back two medium mounts. When they were changed, the ship just just can't focus firepower at all so at a certain point in the game, I ditch it for a more reliable cruiser warship. This is intentional so I'm not going to expect anything.
Capitals
Like Cruisers, I don't have much to complain about here. I don't think there needs to be sweeping changes. I do feel the fuel consumption of Capitals is disproportionate to that of Cruisers. I don't know why an Onslaught chugs 3x the fuel of a Dominator (which is also a hog). The jump from Cruiser to Capital when it comes to travel is immense and while I get you don't want to go joyriding with battleships, its a logistical strain that feels too high.
The only capital I don't jump at a chance to use is the Odyssey. It's just a bit awkward to use and outside of the main battery, it has a ton of small mounts, which I'm not a fan of.
Strangely enough, I'd advocate for the Onslaught to have a single fighter bay, like the battleships of old. Most battleships had a recon plane or small fighter attached at the bow and the Onslaught carrying on that tradition would be interesting.