In the animated feature film, The Nut Job, one of the squirrel characters asks a disheveled appearing rat with if its appearance is the result of mange or bubonic plague. Being always attuned to the mention of infectious diseases in popular culture I notice that rats are often viewed as harbingers of infectious disease, even by animated squirrels.
Rats can be vectors of many infections, most notably plague and murine typhus, in which their fleas can spread plague bacteria to humans. But other less renowned infections can be spread as well.
Interesting pathogens that can be spread via the bite of rat are Streptobacilus monilloformis and Spirillum minus. These bacteria cause "rat bite fever", which has been known to occur for over 2000 years. This disease causes rash and arthritis in the most cases. A select group of cases, however, experience relapsing infection that can lead to endocarditis (infection of the heart valves), meningitis, and sepsis. Untreated, the mortality rate can reach 13%.
Mange, the disease mentioned in the movie, is caused by parasitic mites. Sarcoptic mange is basically the canine version of scabies. Rarely, humans can contract sarcoptic mange, but the affliction is usually self-limited in humans as the mite is unable to complete its lifecycle in humans. Rats have their own specific mites, but can also contract sarcoptic mange.
Overall, though, rats are not in any way special as sources of zoonotic diseases. However, rats will likely continue to be the subject of heightened scrutiny of health inspectors and the public--as the residents of Hamelin can attest.