Running a single-ship fleet and a single-combat-ship fleet are two very different situations, and I can't recommend doing either of them for long. The vanilla game starts you with two combat-capable ships and throws D-mod junkers at you for a very good reason; you really want to have more than one combat-capable ship in 0.8. Nonetheless, it is possible to go it solo.
A single-ship fleet is going to want to run a hybrid ship, something that can carry cargo and fuel and fight off pirates. Something like the Wayfarer, the Mule, or the Apogee. Capitals are right out, they guzzle too much fuel to be run without logistics ships. You'll be limited to doing scanning, salvaging, and surveying, as you won't have the combat power to take bounties. In terms of mods, you'll want to look at DME's combat freighter line and the Tereshkova as well as BRDY's Silverfish, Megaceras, and Hawkmoth.
A single-combat-ship fleet is going to want something fast. Something that can pick its battles, easily get out of being flanked, and that doesn't need support. Ideally something cruiser or capital sized. The very best cruiser for this is the Aurora; Plasma Jets lets it move faster than most frigates for short periods of time, giving it unparalleled mobility in its size class, and it can carry quite a heavy loadout of missiles and energy weapons. Two Heavy Blasters in the central medium energy turrets, two Sabot Pods in the medium synergies, and HE missiles of your choice in the four small synergies will allow it to take out anything smaller than a Paragon as long as you pilot it right. If you want to run a capital, go with either the Legion or the Odyssey. The Legion's burn drive, built-in fighter escort, and wide turret arcs make it exceptionally hard to flank or pin down for a large low-tech ship, and it packs a hell of a lot of firepower. The Odyssey is as fast as a destroyer and with very precise piloting can focus three HEF-boosted large energies on a single target. Mod ships you should consider include but are in no way limited to the DME Zelenograd, Kormaran, and Baikal, the BRDY Kumaraja, the DA Maelstrom, the ORA Elevation, and the SRA Charybdis and Mimir.
All that said, I would highly recommend you try running an Industry/Combat/Technology "garbage ball" fleet instead of going it solo. The Industry skills let you acquire a lot of cheap, substandard, completely disposable junker ships to serve as distractions for your Combat/Technology boosted wündership. It's like going it solo in that you don't have to worry that the AI will throw away valuable ships, but you still have friendlies in the battlespace to flank with.