Müller & Lamprecht has a table of relative frequencies of endgames, but probably not exactly what you want.
I'm trying to get the hang of K+2N v K+Q.
Actually I'm trying to get the hang of K+2N v K+P and this is a sticking point. E.g. I need to be able to answer questions like the following without consulting a tablebase.
In this position (there are many similar) White cannot stop the pawn promoting. So White is not winning. What are his drawing moves? (Two - but that's after consulting a tablebase.)
White to move
Or, which, if any, of these Black to play positions can Black win?
(Black can win the first and last but White can draw the second. After 1...c3 he has just two moves that draw. Again after consulting a tablebase.)
I know that most interesting pawnless endgames (eg not ones like K+R vs K or K+Q vs K which are easy wins or R vs R which is an easy draw) tend to be rare, since it is hard to capture all the pawns, and it usually involves a piece imbalance (and usually if you are up a piece, then you can generally capture some pawns for nothing before it reaches a pawnless endgame). Nevertheless, some pawnless endgames such as K + B + N vs K are also possible if the last piece is sacrificed for the last pawn.
My intuition is that the most common interesting pawn endgames (ones that are not easy wins or draws) are K + Q vs K + R (from a rook endgame where one player sacrifices a rook to promote) and K + R vs K + N (from a rook endgame, that becomes a rook vs pawn endgame after a rook sacrifices itself to stop a pawn, then the pawn underpromotes to a knight to stop checkmate when the attacking king tries to push from behind).
I wonder if there is any source to the most common pawnless endgames and how frequently they appear.