I'm looking for a chess position, hopefully not too contrived, in which it's Black's turn, he has several moves, but each one is refuted by a mate by White.
I know I could take any mate-in-two problem, make White's first move and I'd be left with what I just described, but there's one more criterion: I'd like White's second (mating) move to be different for each of Black's possible moves. Or as many distinct mating moves as possible.
Believe it or not, this is for teaching calculus.
EDIT: Here is another mate-in-two problem with five legal black moves leading to five different checkmates.
[Event "Apprenti Sorcier, 1992"]
[White "ACHBAR Selim"]
[Black "mate in two"]
[Result "1-0"]
[SetUp "1"]
[FEN "8/8/1R6/2p5/8/2Bk1NRN/3P4/K6B w - - 0 1"]
1.Rb1! (1...Ke2 2.Nfg1#)(1...Ke4 2.Nh4#)(1...Kc2 2.Ne1#)(1...Kc4 2.Ne5#)(1...c4 2.Nd4#)
All five different mating moves are done with the same piece, which is very nice.
Just to get this started; here is a mate-in-two problem with three possible black moves and three different checkmates.
[Event "Grand Rapids Herald 1932 (643)"]
[White "WURZBURG Otto B."]
[Black "mate in two"]
[Result "1-0"]
[SetUp "1"]
[FEN "8/Q7/2kp4/8/8/1K6/3R4/5B2 w - - 0 1"]
1.Re2! (1...Kb5 2.Rc2#) (1...d5 2.Re6#)(1...Kd5 2.Bg2#)
I make a separate answer for my own constructions. I'm not a problem composer and don't claim any artistic value.
In the following position black has 18 legal moves, and white has a unique mating move after each black move. The 18 mating moves are all different.
[Title "Black to move"]
[SetUp "1"]
[FEN "kr5R/rb6/8/8/8/4p3/2K3B1/R6Q w - - 0 1"]
So I looked this up in C. Jeremy Morse's Chess Problems: Tasks and Records, as I should have done before entering this fray. The specific task posed by I.J. Kennedy is not addressed, but there are several problems that come close while pursuing some other goal. In particular, a problem by Morse himself (#34, originally in The Problemist 1992), using the same three-lines scheme that Dag Oskar Marsden found independently, is easily modified to attain 21 Black moves each answered by a different mate:
[Title "Mate in 2 (C. Jeremy Morse 1992, adapted NDE 2016)"]
[SetUp "1"]
[FEN "5Q1R/P2p2K1/2bPP1p1/7r/3B1NN1/4p3/8/R5rk w - - 0 1"]
Again two White Queens, one provided by the key move 1 a8Q! One mate appears on each of the 21 squares on the same rank, file, or diagonal as the bKh1.
This is answered in Sir Jeremy Morse's Chess Problems: Tasks and Records, already cited by Professor Elkies. In paragraph 2.4, Morse says "The record for the total number of different White mates (and hence of variations also) in the two-mover is 24, shown in 1, with multiple threats but only a few minor duals." (The problem Morse refers to is the same one in the 1st edition, published 1995, and the 3rd, published in 2016; I show it below.)
If the duals are removed, 24 dual-free lines remain, ending in 24 different mates.
[Title "Nenad Petrovic, The Problemist, 1946. #2"]
[FEN "Q7/7P/3p2p1/2P1R3/1KN2pqp/1P4p1/r3P3/kr5R w - - 0 1"]
1. h8=Q! Ra7 2. Qxa7#
(1... Ra6 2. Qxa6#)
(1... Ra5 2. Qxa5#)
(1... Ra4+ 2. Qxa4#)
(1... Ra3 2. Qxa3#)
(1... Rxh1 2. Qxh1#)
(1... Rxa8 2. Qxa8#)
(1... dxe5 2. Qxe5#)
(1... Qf5 2. Rxf5#)
(1... Qg5 2. Rxg5#)
(1... Qh5 2. Rxh5#)
(1... Qe6 2. Rxe6#)
(1... Qd7 2. Re7#)
(1... Qc8 2. Re8#)
(1... d5 2. Rxd5#)
(1... dxc5+ 2. Rxc5#)
(1... f3 2. Re4#)
(1... Qf3 2. Re3#)
(1... Qxe2 2. Rxe2#)
(1... Rg1 2. Rxg1#)
(1... Rf1 2. Rxf1#)
(1... Re1 2. Rxe1#)
(1... Rd1 2. Rxd1#)
(1... Rc1 2. Rxc1#)
So here we have the idea "line-pinned black line-piece moves away from black king; white pinner captures it" on a rank and on a file, as in Dag Oskar Madsen's and Prof. Elkies's problems, but not also on a diagonal. Instead, in 11 variations, White's other rook is used to discover a diagonal check, and must choose its destination accurately, either to interfere on the line a Black unit threatens to interpose on, or to capture that unit. Black uses a variety of means to make only one square work. wPe2 prevents 1. ... Qd1 and avoids a dual after 1. ... Re1.
Squeezing yet another mate from Dag Oskar Madsen's construction, for a total of 19:
[Title "Black to move"]
[SetUp "1"]
[FEN "R6Q/6B1/2p3K1/8/8/3pN3/rb6/kq5R w - - 0 1"]
How about this mate in two:
1.Nd4!!
1...Bxd4 2.Qb1#
1...Qxd4 2.Qxh7#
1...Kxd4 2.Qb4#
1...exd4 2.Qxd5#
Everything else is answered by: 2.Rg4#
... and yet another mate, for a total of 20, at the cost of a few more White units including a second Queen:
[Title "Black to move (after Dag Oskar Madsen)"]
[SetUp "1"]
[FEN "Q3Q2B/8/4P1K1/1P4p1/2N5/3pN3/rb6/kq5R w - - 0 1"]