The S-Type had two engines during its 5 year run: the 3.4L and 3.8L I6. These were the same engines used in the XK line.
About 25,000 of these cars were produced, with 9,000 being the 420 model. It was supposed to be a replacement for the Mark II, but sales of that car were too strong in 1963 to cut it from Jaguar's lineup, so they simply made both.
The 420 model was in higher demand because the engine was bigger than the standard S-Type model. The 420 got a 4.2L engine. Marles Variomatic power steering and an upgraded electrical system were also offered for the first time with the 420 model.
These cars are obviously not very common on the used car market. I'm sure there are thousands of them left out there, but most are owned by car collectors who rarely sell. You can find them if you look hard enough, and you know where to look. Forget about the normal places I would suggest, like local classifieds or Craigslist. They won't have any. Try eBay or Hemmings. They tend to have a lot more of the classic cars.