To the best to our knowledge, there is only one substantiated case of a car shipped on the Titanic: the 1912 Renault Type CB Coupe de Ville owned by William Carter. This is the same car that featured in Jack and Rose's love scene as well as that one episode of Rick and Morty. This car is known to be on the ship because Carter filed an insurance claim for its loss. There is a possibility that there were other automobiles on the liner, but that is speculation since the Titanic's cargo logs have been lost. There may have been other automobiles on board, but there is no concrete evidence in the surviving manifests or insurance claims.

As to what happened to these cars, well, they went down with the ship. This is pretty common. Stuff that floats on cars such as tires or seat cushions are usually bolted onto stuff that does not. The Hoki Maru of Truk Lagoon is an famous dive spot renowned for its cargo of trucks and the linked video shows how they tend to stay in place. The Titanic's cargo holds were located fairly deep in the bowels of the ship, as you can see in this image. Carter's Renault was very likely crated and other automobiles were likewise probably prepped for transport with crates or secured. So they are likely still among the rusting out wreck of the liner long after their crates have rotted away. The film Ghosts of the Abyss may have caught a glimpse of the Renault, but it is too remote to get a firmer picture.