I often hear people saying what kind of [singular noun] rather than what kind of a [singular noun].
Are we not supposed to use an article (a) before noun?
Thinking in depth about it, either could be considered correct (i.e. neither is positively incorrect), and both are common. It's true that an article should often be used before a noun. Hence the following is incorrect:
It should be:
Or, if you have already been introduced to that person:
However, since phrases like what kind are referring to a type, not an individual, the article a is not mandatory. Consider the phrase what kind of dog. Dog is a type here, not a specific dog.
The question form complicates this problem. Let's start by looking at
This concord suggests that the determiner for kind is bleeding through and filling the determiner slot for X too, which is why the noun in X can be singular without needing a determiner as other singular countable nouns typically do.
Going back to the original question, I think when people say what kind of cow, they take what to fill the obligatory determiner slot for cow and the kind of becomes somewhat transparent. At some level, they are asking what cow. This would be somewhat analogous to the subject-verb agreement in cases like what kind of cows were used (cf. what kind of cows was used).
Those who focus on kind, may be more prone to see the determiner slot for X as unfilled and fill it with a, just as they may force number agreement between kind and was in the example above.
I suppose it depends on the intent of the question.
If you are asking about the type or group of a person, then you may skip the article: "What kind of person dress in sarong to go to work?". Although in this case, I would use people instead of person.
If you are asking about the character of the (specific) person in question, then I think the article is needed, e.g. "What kind of a person who would do such heinous thing to a puppy?"