I believe they're singular nouns. Each of them is the name of a disease, which is one thing.
These diseases have names that are plural in form, but I believe this is a legacy to their origin. "measles" is believed to come from the Dutch "masel", which meant "pustule". So someone with this disease has many of the pustules on their body, and they would have described his affliction by referring to them in the plural: they had many measles.
But over time we've lost most of the singular forms -- we no longer refer to each blister as a "measle". Instead, the plural form has come to mean the affliction itself. They're used the same way that disease names that aren't in a plural form are, such as "leprosy" and "ALS".
The one variety of diseases I can think of where we still have the singular form are the -pox diseases: smallpox, chicken pox. We still use the words "pock" and "pockmark" to refer to the scar left from a blister. But I don't think anyone actually refers to a single blister as a "chicken pock".
Disease names are also normally uncountable, as they describe general concepts. You can't have two measles, although you can have multiple measles outbreaks, patients, epidemics, etc.