I use a horse
But only for my Cleric.
The way horses are in EQ they are not really a means of transport. The cheap ones are slower than a SoW and take longer to stop and start.
If fighting in an outdoor zone then I will use it because it removes sit aggro and means I can cast without having to stand up and sit down. This helps mana regen. If you have just buffed your party at a wayfarer camp and sit down to med only to be interrupted every time a level 10 orc walks past it gets annoying.
I do sometimes ride across hilly places as riding avoids the constant "You have been injured by falling" messages.
I would not use a horse for a melee character. The only exception to that is the Holy/Unholy Steed AA that Paladins and Shadowknights get. Not only is it the fastest horse around but you never lose it - even on CR.
So the answer is that lots of people have them but you have to visit exp camps to see people sitting on them. They do not in general get "ridden into the sunset".