I've had good luck with the high tech ships.
Frigate: Hyperion. 2x pulse lasers, rest of your points into boosting shields & flux dissipation. Really only has issues against very large fleets - and that's only because very large fleets generate maps with lots of objectives, which your one ship can't hold, and then you get run down by swarms of enemy ships moving at high speeds & with extended ranges. Oddly, this should be an easier ship to fly solo once frigates & fighters no longer benefit from nav / sensor nodes.
Destroyer: Medusa. Grav beams, tac lasers, and 2x railgun. Can be tricky going up against high-tech fleets, since you've only got the railguns for generating permanent flux. But absolutely demolishes pirate fleets - you've got a huge range advantage over most stuff, and that many beams will tear through low-tech shields. For a more general purpose build, might be worth considering pulse lasers - though their increased flux generation will reduce your ability to take incoming damage.
Cruiser: Apogee
as described here worked fairly well solo. Aurora is also good; for solo play I tend to go light on the missiles and rely more on the energy weapons; pulse & ir pulse lasers in all the turrets that can aim forward; choice of pd lasers w/ extended range or tac/grav beams for the rear. A harpoon pod in the large missile slot isn't a bad plan, but I often spend the space on more hull mods instead. (If using
my energy balance mod, I'll put burst PD into the rear turrets - but otherwise they're not really worth it.)
Capital: Paragon. Leave the front large slots empty & mount plasma cannons in the large turrets. Flak or dual flak in the universal slots. Grav beams in medium energy, ir pulse in small energy, lots and lots of vents. Takes a bit of care to use well - the flak guns can easily run out of ammo, so you often want to turn them off if you're good on flux and can just take missile hits to shields. Turn them on if you're taking down shields or dealing with bombers - the medium & small energy mounts can easily kill off bombers if you've got flak to punch a hole in their wall of bombs. And, of course, you have to get fairly good at aiming the plasma cannons. For hull mods, you want for sure resistant flux conduits, stabilized shields, and integrated targeting unit; you'll also want at least near max on flux vents.
Of course, any of the larger ships will function a *lot* better with even just a little bit of frigate support; a couple of AI-controlled Tempest or Hyperion frigates will help a lot - send them out to capture nodes, tie up enemy ships so you're not getting totally swarmed, chase down things that are smart enough to run away, etc.
Alternatively, an Odyssey plus a complement of fighters works very nicely - though that's kinda debatable if it really counts as "solo". Probably an ideal setup would use 2x tachyon lances on the left, and a plasma cannon on the right - though 2x plasma and 1x lance is, at least for me, more fun.