My fave is Mustangs! I breed sell and buy them so if you are interested let me know!

https://horsefable.com/hall-of-fame
You can always check this to see the eb and flow of breed popularity!
My favorite is Hanoverians <3

Thank you!

1 being not at all popular and 5 being very popular

Abstang: 2

Akhal-teke: 5

American Azteca: 1.5

American Paint Horse: 5

American Saddlebred: 5

American Warmblood: 5

Andalusian: 5

Anglo Arabian: 3.5

Appalachian Mountain Horse: 4.5

Appaloosa: 5

Appendix Quarter Horse: 3

Arabian: 5

Araloosa: 5

Azteca: 1

Brumbian: 3

Brumby: 4

Chincoteague Pony: 4

Clydesdale: 4

Drum Horse: 2

Dutch Warmblood: 2.5

Friesdale: 3.5

Friesian: 5

Hanoverian: 2

Hispano Arabe: 1

Iberian Warmblood: 1.5

Irish Cob: 5

Irish Draught: 1.5

Irish Sport Horse: 3

Knabstrupper: 3.5

Marwari: 5

Morgan: 4

Mustang: 4.5

National Show Horse: 5

Norwegian Fjord: 3.5

Paso Fino: 4

Percheron: 3.5

Pintaloosa: 3.5

Pony of America: 3.5

Quarab: 1.5

Quarter Horse: 5

Quarter Paint: 2.5

Shetland Pony: 3

Tennesse Walking Horse: 4

Thoroughbred: 5

Walkaloosa: 5

Warlander: 3

Welsh Cob (D): 1

Welsh Mountain Pony (A): 2.5

Welsh Pony (C): 1.5

Welsh Cob Pony Type (B): 1.5

This is in no way 100% accurate. This is just of what I have seen personally based on the rescue, studs/broods, sales, auctions, and people talking about which horses they have. I'd say don't go for any breeds here that I have rated below a 2 if you are new and lack spare (in-game) money to pay for horses not a lot of people are going to buy. If you want you could try raise the populations of some breeds (I'm trying to do that with Abstangs) after you pretty much fully understand the game.

But give it a few months and this will probably be innaccurate.

My favourites are Irish Sport Horse and Brumby.

Are there breeds where on sale it's hard to find horses with good PS or BS? Because if feels like I can never find good thoroughbreds for some reason, or maybe thats just me.

Usually breeds with a wider range of colours (Irish Cob, American Paint, Appaloosa, etc) will have worse stats compared to say Perchons or Friesians who have less colours because people who breed for colours will choose the breeds with a lot of colours. Personally I breed for colour mostly.

TBs however I would say are in the middle.

Basically the more colourful a breed is the worse the stats (on average).