Does anyone know what the record for most mates-in-2 is for a composed problem?
Example: this one has 58 different moves that mate in 2.
[FEN "6R1/8/8/8/3Q3B/5K2/7k/1R6 w - - 0 1"]
Sir Jeremy Morse, in Chess Problems: Tasks and Records, Introduction, cites this construction by E. Luukonen:
[Title "E. Luukonen, Uusi Suomi, 1936"] [fen "kr6/pnPPPPPPPP/P2N4/R7/3BBN2/1Q6/3K3P/6R1 w - - 0 1"] [StartFlipped "0"]
All of White's 117 moves lead to mate in 2. White has no mate in 1, but if Black's rook were to move, or if (after White captures that rook) Black's king were to recapture, White then has a mate. Thus a waiting-move is enough.
However, do note that it is 117 only you count different types of promotions as different moves. If you count promotion as a single move, then the number is 96. The same logic also applies to Glorfindels answer.
I assume you want positions where a mate in one is not possible. Also, these kind of records are often two-fold: one with promoted pieces and one without.
Here is such a position with 76 different mate in twos. I don't think this is anywhere near a record though.
[FEN "8/PPPPPP2/3N2pQ/6Rp/3B2p1/1B4Pk/R6P/4N2K w - - 0 0"]
The main idea is to play a waiting move; Black's only move is h4 after which Qxh4 is mate. This works for all 24 promotions, 10 knight moves, 18 bishop moves, 11 moves by the rook on a2, Kg1, Rxg6, Qh7 and Qh8. On 1. Nf3 and 1. Nf5 Black can respond by taking the knight, but Be6 resp. Qxh5 mates. The moves Rga5, Rb5, Rc5, Rd5, Re5 can also be met with a black g5 (and Rf5 by gxf5), but Qxh5 is mate then. 1. Rxh5 gxh5 2. Qxh5 is mate (also in reverse order). Finally, 1. Rxg4 Kxg4 2. Be6 is also mate.
As Glorfindel noted in his answer, with and without promoted pieces are often stipulations in these sorts of answers. Here’s a try at with promoted pieces that is based of off Rosie F’s given diagram, as the other two answers are for without promoted pieces.
[FEN "kr6/pnQQQQQQ/B2N4/R7/3BBN2/1Q6/3K3Q/6R1 w - - 0 1"]
If my count is right, that is 156. Higher numbers probably exist, but those would require entirely new positions which I do not feel like making.
Another stipulation that could be considered is no promoted pieces with no promotions allowed. Here is my try at it. This should be 71 mates in two if my count is correct.
[FEN "k4N2/2B3Q1/K1P5/p2B1N2/RP6/8/2PPPP2/7R w - - 0 1"]
White has plenty of waiting moves here, and there are no mates in one. Better setups most certainly exist.