Apogee is my overall favorite. It has great utility functions and low maintenance/fuel costs. Armed with Plasma Cannon, two Ion Cannons, Sabot Pod and couple pd lasers it works great against most opponents. It has decent armour even if the shields get shot.
For pure combat ship I like the Aurora best. I equip it with Phase lance, 2 Pulse Lasers, 3 IR Pulse lasers and two Ion cannons, Sabot Pod and one burst PD laser. Lasers and Sabots deal with enemy shields, Phase lance peels off armour and ion cannons quickly turn any enemy into sitting duck. With max vents and flux distributor the ship is almost flux neutral with all weapons firing and hull-modded 360 degree front shields on. I have no plans to move into Capital ship myself. I leave that to more combat skilled AI officers and use the Aurora's great mobility to support them where necessary instead.
My favorite overpowered ship is Drover with Thunder class fighters. With 'Reserve Deployment' Drover can put out roughly the same number of fighters as much more expensive Heron. Drover has convenient missile heavy weapon load-out that lets it dish out major burst damage despite the very weak flux stats, not that it's usually required as the long range Thunders can reach most of the map with Drover staying safely in the rear. Thunders themselves are insanely good deal for 8 ordnance points and will easily cripple even capitals with their ion cannons, not to mention anything smaller.
My favorite frigate is Centurion. It was one of the first ships I added to my fleet and they are still part of my end-game fleet, now being pretty much the only frigate I deploy in major battles. 'Damper Field' makes this already sturdy frigate hilariously tough. They can single-handedly hold several frigates and destroyers at bay when defending tactical points. I don't think I have ever seen anything smaller than a capital deal hull damage to my Centurions. Two light assault guns, two light autocannons and two vulcans with unstable injectors, stabilized shields and armored weapons mounts make these incredible little support ships that still have a place in capital battles where any other frigate would perish in seconds.