Up until events chronicled in Unseen Academicals, public transportation in and around the city of Ankh-Morpork revolved around humans and horses. Depending on length of the journey, there were light hansom cabs prepared to take the paying customer anywhere within the city limits on demand: in Making Money, we note that one such cab took Moist von Lipwig and Miss Drapes from the Royal Bank to Mr Bent's lodgings on Elm Street for a total charge of eleven pence. For longer journeys between cities and countries, there were the the mail coaches, which charged the Wizards fifty dollars a head to go from Ankh-Morpork to Lancre. (Although forty dollars of the fare was a Bandits' Guild surcharge).
Efficient though they are, both these methods of transportation were effectively limited to the middle and upper classes, and, in the main, to humans and the socially better quality Undead and other species, such as Vampires.
Working class humans on low wages would not be as able to pay that eleven pence cab fare, and longer journeys for them, historically, involved walking or hitching lifts on trade barges and carts. (cf Carrot's arrival in the city from Lancre).
Other forms of mass transportation arose to fill the gap, and were seen in more detail in Unseen Academicals. Two forms of cheap local mass transportation first seen in Unseen Academicals were the horse omnibus and the troll chair.
. With the growth of the city and increasing congestion, the appearance of a horse-drawn omnibus is a natural development. Fares are collected by a conductor, all of whom have been dwarfs with a persuasive pickaxe for efficiency. Services currently run to all parts of the city of Ankh-Morpork, with some longer-distance routes running to the nearest neighbouring cities. These rely on efficiency rather than speed, and carry no mail or freight. Fares are cheap because up to eighty people are all going in the same general direction, and over the twenty OR thirty miles of a route there will of course be a continual procession of people getting on, paying short fares, and getting off again. This allows high volume and low unit prices to create the desired level of profit for the operator, and everyone is happy.
The troll chair is rather like the sedan chair on Roundworld, except that in this case, one troll is licenced to carry up to four humans in fairly comfortable enclosed seating slung all over its body, in return for an agreed fare. From the University to Dolly Sisters, for Glenda Sugarbean and Jools, was five pence.
And of course, sedan chairs are still privately used by the nobility all around the city: Samuel Vimes was given one by Vetinari as a wedding present. (He noted it was characteristic of the Patrician to realise he loved walking around the city and to give him a present that made it impossible for him to do so.)
Sedans may also have been used as the equivalent of private taxis.