I had a bit of a frustrating experience with this and after reading the wiki, asking questions in this sub and plenty of trial and error things finally started clicking and I finally found an S class Fighter that makes me smile every time I look at it (boy were there a ton of ugly combinations out there). So I thought I'd share what I learned with others that may be new to the game and are looking to spend their hard earned credits on a fancy spaceship.

1. know your ship costs and come prepared: S class large spaceships will set you back 39 mil. (38/12 explorer) // 58 mil. (38/12 fighter) // 126 mil. (48/8 hauler)
2. know what to look for: look up the NMS wiki or google images of the large ship you want so you know what to look for. Fighters only have 1 specific feature that signifies a large hull, while haulers have multiple for example.
3. know where to look: Any 3 star economy system (PC) will give you the highest chance of getting an S class from the save/reload lottery.
4. know when to give up and move on: Any given system only has 2 large ship designs (ignore shuttles, they don't have large hulls). EDIT: It has been pointed out that there can be more than 2 unique large ships in a system (evidence of 3 in a link below). I've only come across 2 at a time in about 15-20 systems (Euclid galaxy) but I wasn't looking super close at shuttles and explorers, so maybe the limit is 3, but I'm pretty certain there is an actual limit. You can inspect ships with your Visor to check their sizes and rarity. Once you spot 2 unique ship designs with large hulls, if they are not what you're looking for just move on to the next system and try again.
5. system race does not translate into better chance of getting a specific ship type: If you look at the NMS wiki you will find that each race gets a boost to the number of ship designs for a specific class. Gek = extra Haulers, Vy'Keen = extra fighters, Kovax = extra explorers. This does not in any way influence which of those ships will have large hulls. 2 are chosen at random and in my experience they are typically not the system preferred class type. I spent days combing through Vy'keen systems looking for a decent fighter with no luck mainly because there were no large fighters there, only to switch to a Gek system and immediately spotted the perfect fighter for me with a large hull.
6. quickest way to check all available ship designs in a system: Use your ship's economy scanner (make sure to install) to find the trade outpost in that system. Land there and use your jetpack to reach the top of the building. Drop your save beacon and do a manual save. Reload the manual save and watch the sky with your Visor on. Once you've counted 7 (sometimes 8) ships flying about and/or landing around you, if there is nothing you like you can reload the save and try again. Once you've spotted 2 different large hull designs (30+ slots // 40+ for haulers) you can either: A) leave the system because you don't like the designs/colors or B) if you spotted a ship you want, continue to reload over and over until the ship you want shows up as an S class (or A with maxed slots if you don't care about class bonuses). Know that with option B) you're now playing the lottery and if you want that perfect S class variant you could spend many many many hours reloading over and over till it shows up. Don't lose hope it'll show up, thought chances are you might spot a few exotics before you see that S class.

Other tips: When it comes to the different ship types, know that haulers only give you 6 extra slots for an extra 60+ million credits. At some point in the game, money becomes irrelevant so it's good to consider the other differences, such as maneuverability, damage potential, shield strength and hyper-drive distance. You may also want to ask yourself "do I need 1 ship that can do it all, or can I have different ships for different purposes". For example if you want a ship to explore new systems with, any S class explorer regardless of size will do the job just fine. I maxed out a mid-sized S explorer with hyper-drive modules and now I can travel vast distances. Once I'm in a new system that I want to further explore, I call up my freighter and swap to my fighter which has no hyper-drive modules but max shield/pulse drive/weapons of choice with plenty of slots left for cargo. Also, don't forget to "test-drive" all the various ship designs to make sure you like the cockpit and/or the ships engines as you'll be looking at those quite a bit. I came across a large S class hauler that I seriously considered buying so I did a manual save, bought it, jumped in and immediately hated the inside of the cockpit as my view was reduced to 20% of my screen size. The 3rd person view wasn't that great either, so this was an easy decision for me.

Oh, and don't forget your Freighter can only fit 6 ships, so don't forget to switch to a throw-away ship you can trade in before going on a hunt for a new ship.

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That's all I can think of right now. Now to figure out the Multi-tool lottery….

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Happy spaceship hunting, interlopers 🙂

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EDIT: People have pointed out that the limit for large hulls per system is greater than 2 so I've updated point 4 to reflect that. Thanks for the info.