According to the BBC English leaners website and The English-plus University of Toronto site the following rules apply:
- We use kind of if the noun is uncountable or a singular countable noun.
Ex What kind of coffee is that? coffee= uncountable
Ex. What kind of apple are you eating? apple= countable/singular
- If the noun is plural then we use kinds of
Ex. I like all kinds of animals. animals = plural
Another way of saying all of this is that if the object of the preposition of is singular or uncountable then us kind.
If the object of the preposition of is plural use kinds of
Also, if we use quantifiers like all, several, some , many then we always follow them with kinds of
Ex. I like all kinds of cars.
Finally, according to most dictionaries kind of, type of and sort of all have the same meaning =type or variety of
I hope that helps.