2007, in Romania [1][2]. More specifically, horse-drawn carts were banned. From what I can see, the movement to ban them stems from an argument that horse-drawn carts/carriages are a form of animal cruelty [3]. I haven't seen anything that singles out/bans horses (with riders, rather than carriages) though.

[1] https://www.forbes.com/sites/nadiaarumugam/2013/02/14/the-ro...

[2] http://www.telegraph.co.uk/news/worldnews/1578965/Horses-lef...

[3] https://www.peta.org/issues/animals-in-entertainment/horse-d...

Depends on your location. In my country they are banned on highways, national roads and cities or in other words in all those places in which cars rule.

Well, in the Netherlands and I think most other European countries, they are banned on car-only roads (i.e., motorways and expressways; autosnelweg and autoweg in Dutch). Elsewhere, including cities, they are allowed.